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This is a resource I found that is probably the most creative idea that I have seen in a long time. This teacher presented Algebra Notes in the form of newsletters. Awesome idea!

This is a resource I found that is probably the most creative idea that I have seen in a long time. This teacher presented Algebra Notes in the form of newsletters. Awesome idea!

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- 1. Algebra Newsletter NotesA creative way to presentconcepts, notes & study guidesin a newsletter format.Excellent Resource!www.clubtnt.org/my_algebra
- 2. In many ways, working withrational expressions is like workingwith fractions. Recall that:, equalnotdoesbbcabcba +=+0, equalnotdoesbbcabcba −=−As long as we have commondenominators, adding andsubtracting rational expressions isfairly straightforward – simplyrewrite the addition or subtractionon the common denominator.0equalnotdoesRRQPRQRPRQPRQRP−=−+=+To add or subtract rationalexpressions with differentdenominators, we need to find theLowest Common Denominator forthe polynomials that represent thedenominators or our rationalexpressions.To find the LCD forpolynomials:1. Factor each denominatorcompletely. Use exponentnotation for repeated factors.2. Write the product of all thedifferent factors that appear inthe denominators.3. On each factor, use the highestpower that appears on thatfactor in any or thedenominators.Example: LCD of 20, 501. Factor completely: 20=22*5,50=2*522. Product of all different factors:2 and 53. Use highest power: 22*52=100Example: LCD x3yz2, x5y2z, xyz51. Factor completely, eachexpression already is a factor:x3yz2, x5y2z, xyz5.2. Product of all the differentfactors (x, y, z)3. Use highest power: x5y2z5Example: LCD a2+5a+6, a2+4a+41. Factor completely: (a+2)(a+3)and (a+2)22. Product of all different factors:(a+2), (a+3), and (a+2)3. Use highest power:(a+3)(a+2)2Adding and Subtracting RationalExpressions with DifferentDenominatorsWhen we add or subtractrational expressions with differentdenominators, we must convertthem to contain identicaldenominators. The easiest way todo this is to use the LCD that wehave just reviewed. Once the LCDis determined, rewrite each rationalexpression as an equivalent rationalexpression with the LCD. Theprocess is just like working withfractions.Example:127203+1. Factor completely: 20=22*5,50=2*522. Product of all different factors:2 and 53. Use highest power: 22*52=100151115*411*4604460356095512733203===+=+=Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Mr. Breitsprecher’s Edition April 5, 2005 Web: www.clubtnt.org/my_algebra
- 3. Mr. Breitsprecher’s Edition April 5, 2005 Algebra Connections, Page 2Example:15461−1. Factor completely: 6=2*3,15=3*52. Product of all different factors:2, 3, and 53. Use highest power: 2*3*5=301013*103*1303308305221545561−=−=−=−=−Rational expressions are addedand subtracted just like rationalnumbers (quotients of rationalnumber or fractions). Whennecessary, rewrite rationalexpressions with a commondenominator.Follow these steps to add orsubtract rational expressions whenthe denominators differ.1. Find the Lowest CommonDenominator (LCD)2. Rewrite each rationalexpression as an equivalentexpression whose denominatoris the LCD.3. Add or subtract numerators andplace the sum or difference overthe common denominator.4. Write the result in lowest terms.Example:3225+x+=xxx 22323325xx6415 +=Example:xxx 329122++−)3(2)3)(3(1+++−=xxxx)3)(3(63)3)(3(6233)3(2)3)(3(1+−−=+−−+=−−+++−=xxxxxxxxxxxxxxxxxExample:5254−−− aaNote: because (5-a) and (a-5) differ onlyin signs, we can obtain identicaldenominators by multiplying only the firstexpression by -1 in BOTH the numeratorand denominator (maintain and equivalentexpression).565652545211454−−=−−=−−−−=−−−−−=aaaaaExample:6112612212−+++++xxxxxNote: The LCD for these threedenominators is 6x(x+2)xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx63)2(6)2)(3()2(6652)2(6222266)2(622)2(622)2(666)2()2(61)2(61266)2(161)2(612)2(1+=+++=+++=+−−+++=++−+++++=++−+++++=−+++++=Online ResourcesFinding the Lowest CommonDenominatorhttp://www.csun.edu/~ayk38384/Math093-Rational%20Expression.htmhttp://library.thinkquest.org/20991/textonly/alg/frac.htmlAdding and Subtracting RationalExpressionshttp://www.mathsteacher.com.au/year10/ch14_rational/05_addition_and_subtraction/addsub.htmhttp://a-s.clayton.edu/garrison/Math%200099/sect44.htmhttp://faculty.ed.umuc.edu/~swalsh/Math%20Articles/RationalE.htmlhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut33_addrat.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut33_addrat.htmhttp://www.algebra-online.com/adding-substracting-rational-expressions-like-denominators-1.htmhttp://tutorial.math.lamar.edu/AllBrowsers/1314/RationalExpressions.asphttp://www.sparknotes.com/math/algebra2/rationalexpressions/section2.rhtmlPurple Math’s Tutorial: Addingand Subtracting RationalExpressions (2 parts)http://www.purplemath.com/modules/rtnladd.htmhttp://www.purplemath.com/modules/rtnladd2.htmFactoring a Polynomial:1. Are there any common factors? If so,factor them out.2. How many terms are in thepolynomial:a. Two terms: Is it the difference of2 squares? a2-b2= (a-b)(a+b)3. See if any factors can befactored further. Watch fordifference of 2 squares!Quick Review: Two ImportantExponent Rulesam*an= am+nam/an= am-na. Three terms: Try one of thefollowing patternsi. a2+2ab+b2= (a+b)2ii. a2-2ab+b2= (a-b)2iii. Otherwise, try to use anothermethod.b. Four terms: Try factoring bygrouping.
- 4. Mr. Breitsprecher’s Edition January 18, 2005 FREE!Decimals are a method ofwriting fractional numbers withoutwriting a fraction having anumerator and denominator.The fraction 7/10 could bewritten as the decimal 0.7. Theperiod or decimal point indicatesthat this is a decimal.If a decimal is less than 1, placea zero before the decimal point.Write 0.7 not .7There are other decimals suchas hundredths or thousandths. Theyall are based on the number ten justlike our number system. A decimalmay be greater than one, 3.7 is anexample.The fraction 37/100 could bewritten as the decimal 0.37. Theperiod or decimal point indicatesthat this is a decimal. Of course, adecimal with one hundredthsdecimal may be greater than 1.Decimal numbers, such asO.6495, have four digits after thedecimal point. Each digit is adifferent place value.The first digit after the decimalpoint is called the tenths place value.There are six tenths in the numberO.6495.The second digit tells you howmany hundredths there are in thenumber. The number O.6495 hasfour hundredths.The third digit is thethousandths place.The fourth digit is the tenthousandths place which is five inthis example.Therefore, there are six tenths,four hundredths, nine thousandths,and five ten thousandths in thenumber 0.6495.If there are two decimalnumbers we can compare them.One number is either greater than,less than or equal to the othernumber.A decimal number is just afractional number. Comparing 0.7and 0.07 is clearer if we compared7/10 to 7/100. The fraction 7/10 isequivalent to 70/100 which isclearly larger than 7/100.Therefore, when decimals arecompared start with tenths place andthen hundredths place, etc. If onedecimal has a higher number in thetenths place then it is larger than adecimal with fewer tenths. If thetenths are equal compare thehundredths, then the thousandths etc.until one decimal is larger or there areno more places to compare. If eachdecimal place value is the same thenthe decimals are equal.Sound simple – great! This is thefoundation of percentages, a veryimportant tool for analysis.Source: www.aaamath.comTo write decimals as fractions,use the “placeholder” as thedenominator and write the number(without the decimal) as thenumerator.To write fractions as decimals,divide the numerator by thedenominator.To write a percent as a decimal,drop the % symbol and move thedecimal points two places to the left.To write a decimal as a percent,move the decimal point two places tothe right and attach the % symbol.To write fraction as a percent,write the decimal as a fraction(above), perform that division, andmultiply by 100.To Add or Subtract Decimals1. Write the decimals so thatthe decimal points line upvertically.2. Add or subtract for wholenumbers.3. Place the decimal point in thesum or difference so that it linesup vertically with the others.To Multiply Decimals1. Multiply the decimals as thoughthey are whole numbers.2. The decimal point in the productis placed so that the number ofdecimal places is equal to theSUM of the number of decimalplaces in the factors.To Divide Decimals1. Move the decimal point in thedivisor to the right until thedivisor is a whole number.2. Move the decimal point in thedividend to the right the SAMENUMBER OF PLACES as thedecimal point was moved in step1.3. Divide. The decimal point in thequotient is directly over themoved decimal point in thedividend
- 5. Widgets by Plant As FractionFraction As% of Whole As % of TotalPant 1 342.166 342 1/6 16.66666667 24.64449928Plant 2 30.9 30 9/10Plant 3 7.9558 7 65/68Plant 4 61.619Plant 5 0.13209Plant 6 38.87Plant 7 45.4Plant 8 75Plant 9 37.39Plant 10 94.728Plant 11 56.2Plant 12 57.16Plant 13 6.251Plant 14 52.6Plant 15 32.14Plant 16 99.34Plant 17 26.2Plant 18 99.3302Plant 19 15.229Plant 20 53.7Plant 21 8.268Plant 22 44.5Plant 23 56.5801Plant 24 46.748Total 1388.40719What’s A Widget?A “widget” is an imaginary unit ofproduction – it is often used in Economicclasses to illustrate ideas (it is easy to createdata for imaginary units – virtually anythingworks!)Please create the spreadsheet on the left,using ALL HEADINGS including PLANTNumbers. ONLY ENTER THE DATA FORTHE UNITS OF PRODUCTION (secondcolumn) when you start. We will enter theinformation under: As a Fraction, Fraction as% of Whole, as % of Total, and Total later byapplying our Decimals & Fraction Review,using formulas or functions as appropriate.Please apply BOLD as illustrated to theleft. If you have any problems entering thenumbers, it is usually because EXCEL hasdefaulted to one format (for example, DATE),when we are looking for another format (i.e.number). ONLY USE A “GENERAL”NUMBER FORMAT.If you have any questions or problems,PLEASE ASK A CLASSMATE OR YOURINSTRUCTOR. This activity is written toassume that students have worked with EXCELenough that specific directions about thesoftware are not needed. Please do not feel shyor hesitate to ask for assistance if necessary.PROCEDUREOnce you have entered the information per the instructionsunder the section, WHAT’S A WIDGET, let’s review ourconcepts from our Review Chapter, Decimals and Percents.Please follow the directions carefully and do not use any otherformats, functions, or MS EXCEL features. You will Emailthis to your instructor and no credit will be given for projectsthat have not applied the review principals in this project.1. The column, AS A FRACTION will be used to re-writethe decimal portion (partial unit) of each plant’sproduction as a fraction (including the whole number ofunits). Simply RE-ENTER THE WHOLE NUMBERFROM EACH DECIMAL and then ENTER A SPACE.2. Express the partial unit, after that space, by RE-ENTERING THE DECIMAL PART OF THENUMBER (without the decimal), FOLLOWED BY /(for the fraction bar) and then the DECIMALPLACEHOLDER (10, 100, 1000, or 10000).3. Note how EXCEL will re-calculate the fraction in itssimplest term. (Note examples above). For this class,ALL FRACTIONS IN FINAL ANSWERS MUST BESIMPLIFIED!4. Use AUTOSUM (or SUM FUNCTION) to total the number ofwidgets for ALL plants (cell B26 in my example, if you havedifferent cells, adjust accordingly).5. Next, in COLUMN C, create a formula that will convert thefraction in COLUMN B (DO NOT INCLUDE THE WHOLENUMBER) to a PERCENT OF WHOLE. Note that fromour review, this involves writing the decimal as a fraction,performing the indicated division, and the results multiply by100. Please also note that we always started with a fraction thatwas less than 1 – our calculated percentage, therefore, has to beless than 100. (In example shown, =1/6*100)6. Next, we will calculate each plants output as a percentage of thetotal. Note that each plant only represents a small fraction ofthe total. The percentage of each plant’s output to the totalMUST BE LESS THAN 100. The sum of these percentagesMUST EQUAL 100. Therefore, we will create a formula thatwill divide each plants output by the total of all plants. Tosimplify things, use an ABSOLUTE CELL REFERENCEand FILL this formula down. In this example, I used=B2/$B$26*1007. Use AUTOSUM to verify that the percentages in this columntotal to 100 and add the three borders that are shown in theexample. Be sure all headings and data are displayed.8. Save your work to your stustorage account. Your instructorwill give directions to Email this assignment in to earn credit.
- 6. howMr. Breitsprecher’s Edition March 9, 2005 Web: www.clubtnt.org/my_algebraEasy does it! Lets look at howto divide polynomials by startingwith the simplest case, dividing amonomial by a monomial. This isreally just an application of theQuotient Rule that was coveredwhen we reviewed exponents.Next, well look at dividing apolynomial by a monomial. Lastly,we will see how the same conceptsare used to divide a polynomial by apolynomial. Are you ready? Let’sbegin!Quotient of a Monomial by aMonomialTo divide a monomial by amonomial, divide numericalcoefficient by numerical coefficient.Divide powers of same variableusing the Quotient Rule ofExponents – when dividingexponentials with the same base wesubtract the exponent on thedenominator from the exponent onthe numerator to obtain the exponenton the answer.Examples:• (35x³ )/(7x) = 5x²• (16x² y² )/(8xy² ) = 2xRemember: Any nonzero numberdivided by itself is one (y²/y² = 1),and any nonzero number to the zeropower is defined to be one (ZeroExponent Rule).• 42x/(7x³ ) = 6/x²Remember: The fraction x/x³simplifies to 1/x². The NegativeExponent Rule says that anynonzero number to a negativeexponent is defined to be onedivided by the nonzero number tothe positive exponent obtained. Wecould write that answer as 6x-2.Quotient of a Polynomial by aMonomialTo divide a polynomial by amonomial, divide each of the termsof the polynomial by a monomial.This is really an example of how weadd fractions, recall:cbcacba+=+Think of dividing a polynomial by amonomial as rewriting each term ofthe polynomial over thedenominator, just like we did whenworking with fractions. Then, wesimplify each term.Example:• (16x³ - 12x² + 4x)/(2x)= (16x³)/(2x)-(12x² )/(2x)+(4x)/(2x)= 8x² - 6x + 2Review: Exponents and Polynomialsanmeans the product of n factors, each of which is a (i.e. 32= 3*3, or 9)Exponent Rules• Product Role: am*an=am+n• Power Rule: (am)n=amn• Power of a Product Rule: (ab)n= anbn• Power of a Quotient Rule: (a/b)n=an/bn• Quotient Rule: am/an=am-n• Zero Exponent: a0=1, a ≠0Working with Polynomials• Adding Polynomials. Remove parenthesis and combine like terms.• Subtracting Two Polynomials. Change the signs of the terms in the secondpolynomial, remove parenthesis, and combine like terms.• Multiply Polynomials. Multiply EACH term of one polynomial by EACHterm of the other polynomial, and then combine like terms.Special Products• FOIL When multiplying binomials, multiply EACH term in the firstbinomial by EACH term in the second binomial. This multiplication ofterms results in the pattern of First, Outside, Inside, and Last.• Difference Of Squares: (a+b)(a-b) = a2-b2• Perfect Square Trinomials:(a+b)2= a2+2ab+b2(a-b)2= a2– 2ab + b2
- 7. Quotient of a Polynomial by aPolynomialTo divide a polynomial by a polynomial,use long division, similar to the longdivision technique used in arithmetic.Remember in starting the long divisionprocess:1. Write dividend and divisor in termsof descending powers of variable,leaving space for any missingpowers of the variable or writing inthe missing powers with coefficientzero. If there is more than onevariable, arrange dividend anddivisor in terms of descending order(from the term with the highest“degree” to the lowest “degree”).2. Divide first term of divisor into firstterm of dividend (On subsequentiterations, into first term of previousdifference). Place this answer abovelong division symbol.3. Multiply divisor by the expressionjust written above division symboland align like terms.4. Subtract line just written from lineimmediately above it. Remember tosubtract we change the sign of theMr. Breitsprecher’s Edition March 9, 2005 Algebra Connections, Page 2subtrahend and add (add theopposite).5. Repeat steps 2 through 4 untilthe difference you obtain is apolynomial of degree less thanthe degree of the divisor.6. If the final difference is zero,the division is exact. Thequotient is the polynomial givenacross the top. If the differencein nonzero division is not exact,the quotient is the polynomialgiven across the top plus theremainder (polynomial in lastline) divided by the divisor.Example:)12/()1(32450412223−−++−+++xRxxxxxx2324 xx −xx −223646−−xx1−Source: http://home.sprynet.comAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.OnlineResourcesHow to Add, Subtract, Multiply,and Divide Polynomialshttp://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Polynomial.htmlDividing Polynomialshttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut35_div.htmhttp://tutorial.math.lamar.edu/AllBrowsers/1314/DividingPolynomials.aspDr. Math: Dividing Polynomials.http://mathforum.org/library/drmath/view/52901.htmlLong Division of Polynomials.http://www.sosmath.com/algebra/factor/fac01/fac01.htmlhttp://www.purplemath.com/modules/polydiv2.htmhttp://www.mathwords.com/p/polynomial_long_division.htmhttp://www.math.utah.edu/online/1010/euclid/http://tutorial.math.lamar.edu/AllBrowsers/1314/DividingPolynomials.aspPolynomial Division andFactoringhttp://campus.northpark.edu/math/PreCalculus/Algebraic/Polynomial/FactoringOnline Solutions. Scroll down thissite to the form identified asDivision of polynomials: p(x) /f(x). It will divide polynomials todegree 6. Just set the coefficients,and click "Divide". Be sure that thedegree of p(x) >= degree of f(x).Leading zero coefficients areignored – use zeros for terms in theform that are not present in theproblem you want to solve.http://www.egwald.com/linearalgebra/polynomials.phpDefinitionsMonomial has one term: 5y or -8x2or 3.Binomial has two terms: -3x2+ 2, or 9y- 2y2Trinomial has 3 terms: -3x2+ 2 +3x, or9y - 2y2+ yDegree Of The Term is the exponent ofthe variable: 3x2has a degree of 2.Degree of a Polynomial is the highestdegree of any of its terms.When the variable does not have anexponent -understand that theres a 1.One thing you will do whensolving polynomials is combine liketerms. Understanding this is the keyto accurately working withpolynomials. Let’s look at someexamples:• Like terms: 6x + 3x - 3x• NOT like terms: 6xy + 2x - 4The first two terms are like and theycan be combined:5x2+ 2x2– 3Combining like terms, we get:7x2– 3
- 8. There are several methods fortesting the divisibility of a numberwithout actually performing thedivision. A great interactive tutorial isat:http://www.oswego.org/mtestprep/math8/a/divisibilityrulesl.cfm (interactivetutorial)Dividing by 2• All even numbers are divisibleby 2. E.g., all numbers ending in0,2,4,6 or 8.Dividing by 3• Add up all the digits in the number.• Find out what the sum is. If thesum is divisible by 3, so is thenumber• For example: 12123(1+2+1+2+3=9) 9 is divisible by 3,therefore 12123 is too!Dividing by 4• Are the last two digits in yournumber divisible by 4?• If so, the number is too!• For example: 358912 ends in 12which is divisible by 4, thus so is358912.Dividing by 5• Numbers ending in a 5 or a 0 arealways divisible by 5.Dividing by 6• If the Number is divisible by 2 and 3 itis divisible by 6 also.Dividing by 7 (2 Tests)• Take the last digit in a number.• Double and subtract the last digit inyour number from the rest of the digits.• Repeat the process for larger numbers.• Example: 357 (Double the 7 to get 14.Subtract 14 from 35 to get 21 which isdivisible by 7 and we can now say that357 is divisible by 7.OR• Take the number and multiply eachdigit beginning on the right hand side(ones) by 1, 3, 2, 6, 4, 5. Repeat thissequence as necessary• Add the products.• If the sum is divisible by 7 - so is yournumber.• Example: Is 2016 divisible by 7?• 6(1) + 1(3) + 0(2) + 2(6) = 21• 21 is divisible by 7 and we can now saythat 2016 is also divisible by 7.Dividing by 8• This ones not as easy, if the last 3digits are divisible by 8, so is the entirenumber.• Example: 6008 - The last 3 digits aredivisible by one, therefore, so is 6008.Dividing by 9• Almost the same rule and dividing by 3.Add up all the digits in the number.• Find out what the sum is. If the sum isdivisible by 9, so is the number.• For example: 43785 (4+3+7+8+5=27)27 is divisible by 9, therefore 43785 istoo!Dividing by 10• If the number ends in a 0, it is divisibleby 10.Seem like too much? Easy does it –work on mastering the rules for the primenumbers: 2, 3, 5, and 7 (Note: 7’s arecommon football scores).1. A scholarship fund has $213,198.Can the money be awarded equallyamong 8 students?2. 2465 girls signed up to live in adormitory. If each room can hold 3girls, will each room be completelyfilled after all the girls have beenassigned a room?3. Nine teachers receive 190 filefolders. Can each teacher have thesame number of folders?4. Professor Fields spilled in on amemo from the math department.Find the missing Digit:TO: All Math Faculty9 classes have been scheduled forthis year. Each class has an equalnumber of students. The totalnumber of students is 4__25.5. There are 128 students enrolled in amath course. Can the students bedivided into 4 equal classrooms?6. The headwaiter at the restaurantforgot the last digit of the numberof people in a banquet party. Theparty will be seated in tables of 9.Find the missing digit. It will be aparty of 167__7. The English department has 190pens. Can 8 teachers receive thesame number of pens?8. The school health center has 124boxes of gauze. Can each of the 4wards receive an equal number ofboxes?9. A table seats 4 people. Can 1,5024people sit at these tables and havethem all be full?10. Five students washed cars all daySaturday. They made $108.00.Can they divide this amountevenly?11. Could three students divide$108.00 evenly?12. Could 4 students divide $108.00evenly?Mr. Breitsprecher’s Edition January 20, 2005 Web: www.clubtnt.org/my_algebra
- 9. 7272727Working with exponents is animportant part of algebra and otherclasses. Lets go over theprocedures of exponent rules indetail and review some examples.Rules of 1There are two simple "rules of1" to remember.First, any number raised to thepower of "one" equals itself. Thismakes sense, because the powershows how many times the base ismultiplied by itself. If its onlymultiplied one time, then its logicalthat it equals itself.Secondly, one raised to anypower is one. This, too, is logical,because one times one times one, asmany times as you multiply it, isalways equal to one.x1= x31= 31m= 114= 1 * 1 * 1 * 1 = 1Product RuleThe exponent "product rule"tells us that, when multiplying twopowers that have the same base, youcan add the exponents. In thisexample, you can see how it works.Adding the exponents is just a shortcut!xm* xn= xm+n42* 43= 4 * 4 * 4 * 4 * 442+5= 45Power RuleThe "power rule" tells us that toraise a power to a power, justmultiply the exponents. Here yousee that 52raised to the 3rd power isequal to 56.(xm)n= xm*n(52)3= 56Note: This is really where thePower of a Quotient rule comesfrom. Numbers without exponentsare assumed to be to the power of 1.(a/b)m= am/bm, b ≠ 0Quotient RuleThe quotient rule tells us thatwe can divide two powers with thesame base by subtracting theexponents. You can see why thisworks if you study the exampleshown.xm÷ xn= xm-n, x ≠ 045÷ 42= 4 * 4 * 4 * 4 * 4 / 4 * 44 5-2= 4 3Zero RuleAccording to the "zero rule," anynonzero number raised to the powerof zero equals 1.x0= 1, x ≠ 0Negative ExponentsThe last rule in this lesson tells usthat any nonzero number raised to anegative power equals its reciprocalraised to the opposite positivepower.x-n= 1/xn4-2= 1/42= 1/16Source: www.math.comMr. Breitsprecher’s Edition October 4, 2005 Web: www.clubtnt.org/my_algebraCommon Errors With Exponent RulesWith practice, we can all apply and work with these rules. Sometimes, we learn howto do things by making mistakes. Here are some common errors with exponents.Please review the following examples of how NOT to work with exponents.1. The exponent next to a number applies ONLY to that number unless there areparenthesis (grouping) that indicate another number or sign is actually part ofthe base.-52≠ (-5)2-(5)(5) ≠ (-5)(-5)-25 ≠ 252. The product rule (xm* xn= xm+n) only applies to expressions with the samebase.42* 23≠ 82+3(4)(4)(2)(2)(2) ≠ 85128 ≠ 32,7683. The product rule ((xm* xn= xm+n) applies to the product, not the sum of 2numbers22+ 23≠ 22+3(2)(2) + (2)(2)(2) ≠ 254+8 ≠ (2)(2)(2)(2)(2)12 ≠ 32Math.com has a variety of online lessons and interactive tutorials to help studentsmaster many of the concepts in our math. Check out the “workout” of 10 interactiveexponent examples with solution and helpful tips at:http://www.math.com/school/subject2/practice/S2U2L2/S2U2L2Pract.html
- 10. Everything we have done up tothis point in Beginning Algebra hasbeen to get ready to apply basicprocedures to more involvedalgebraic concepts such as factoringpolynomials.As a quick review, let’s startout by reviewing what factoring is.Factoring is a process to determinewhat we can multiply to get thegiven quantity. If means to write anas a product – the reverse ofmultiplication. In this example, wewant to rewrite a polynomial as aproduct.Examples: Factors of 12Possible Solutions: 2*6 or 3*4or 2*2*3 or [1/2(12)] or (-2*-6) or(-2*2*-3) Note: There are manymore possible ways to factor 12, butthese are representative of many ofthem.A useful method of factoringnumbers is to completely factorthem into positive prime factors. Aprime number is a number whoseonly positive factors are 1 and itself.For example 2, 3, 5, and 7 are allexamples of prime numbers.Examples of numbers that aren’tprime include 4, 6, and 12.If we completely factor anumber into positive prime factors,there will only be one solution.This is why prime factorization isuseful. That is the reason forfactoring things in this way. For ourexample above with, the completefactorization of 12 is: (2*2*3).Factoring PolynomialsFactoring polynomials issimilar, determine all the terms thatwere multiplied together to get theoriginal polynomial. For many of us,it is probably easiest to factor insteps – start with the first factors wesee and continue until we can’t factoranymore. When we cannot find anymore factors we will say that thepolynomial is completely factored.Example: x2-16 = (x+4)(x-4).This is completely factored, wecannot find another way that thefactors on the right can be furtherfactored (think: write as product).Example: x4-16 = (x2+4)(x2-4).This is not completely factored,because the second factor on theright can be further factored. Do yousee that it is one of our specialproducts, a difference of squares?Breaking that example down further,we see: x4-16 = (x2+4)(x+2)(x-2).Greatest Common FactorOne way to factor polynomials,probably the easiest one whenapplicable, is to factor out thegreatest common factor. In generalthis should ALWAYS be the firstmethod we consider wheneverfactoring.Start by looking at each termsand determine if there is a factorthat is in common to all the terms.If there is, factor it out of thepolynomial. Let’s take a look atsome examples.Example: 8x4-4x3-10x2Notice that we can factor out a2 from every term, we can alsofactor out an x2. Some of us willsee that we can factor out a 2x2immediately, but we can also factorthis in steps if that helps us getstarted. Our solution to thisproblem is: 2x2(4x2-2x+5).Example: x3y2+3x4y+5x5y3On closer inspection, we cansee that each term has a commonfactor has x3y. Of course, we mighthave seen this in steps, that eachterm has a factor of x3and then thateach has a factor of y. It doesn’tmatter, as long as we keep lookingfor factors. Our solution is:x3y(y+3x+5x2y2)Example: 3x6-6x2+3xDo you see that each term has acommon factor (greatest commonfactor) of 3x? Please remember,when we factor a 3x out of the lastterm (3x), we are left with +1. BeAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Mr. Breitsprecher’s Edition March 10, 2005 Web: www.clubtnt.org/my_algebra
- 11. Mr. Breitsprecher’s Edition March 10, 2005 Algebra Connections, Page 2careful, forgetting this 1 is acommon error when factoring! Oursolution is: 3x(x5-2x+1)Example: 9x2(2x+7)-12x(2x+7)At first glance, this one looksstrange. It is factored like the otherexamples, however. There is a 3x ineach term and there is also a (2x+7)in each term. Both can be factoredout of this polynomial. This leavesa 3x in the first term (because3x*3x=9x2) and a -4 in the second(3x*-4 gives us our original -12x).The solution is: 3x(2x+7)(3x-4)Factoring By GroupingThis is another way to factorpolynomials that have 4 terms. Itwill not work with every 4 termpolynomial, but when it can be used,it is a simple and direct approach. Itis also a great review of important,basic algebraic procedures. Here isa step-by-step procedure forfactoring 4 term polynomials bygrouping:1. Group terms into 2 groups of 2terms.2. Factor out the greatest commonfactor from each of these twogroups.3. If we are left with a commonbinomial factor, factor it out –then we are done!4. If not, rearrange the terms andtry steps 1-3 again.Example: 3x2-2x+12x-81. Group terms into 2 groups of 2terms: (3x2-2x) + (12x-8)2. Factor out the greatest commonfactor from each term:x(3x-2) + 4(3x-2)3. Factor out a common binomialfactor. Our solution is:(3x-2)(x+4).That’s it! We do not need todo step 4 because we had a commonbinomial term of (3x-2). Once wefactor it out, we are done.Example: x4+x-2x3-21. Group terms into 2 groups of 2terms. BE CAREFUL WITHPOLYNOMIALS WITH A “-” SIGN IN FRONT OF THE3RDTERM! The process is thesame, but notice that we have acommon factor in the 3rdand 4thterms of -1 or just “-”. If wefactor out this “-” and group thefirst 2 and last 2 terms together,we get: (x4+x)-(2x3+2). THISIS AN IMPORTANT STEPWHEN WE HAVE A “-”SIGN IN THE THIRDTERM.2. Factor out the greatest commonfactor from each term:x(x3+1)-2(x+1)3. Factor out a common binomialfactor. Our solution is:(x3+1)(x-2).Example: x5-3x3-2x2+61. Group terms into 2 groups of 2terms. Again, NOTICE THATWE HAVE A “-” IN FRONTOF THE THIRD TERM.Also note the “+” in front of the4thterm. We will still factor outthe ‘-’, so that we do not losetrack of it. This gives us:(x5-3x3) – (2x2-6).2. Factor out the greatest commonfactor from each term:x3(x2-3)-2(x2-3)3. Factor out a common binomialfactor. Our solution is:(x2-3)(x3-2).Example: 5x-10+x3-x21. Group terms into 2 groups of 2terms. Note that in this case,the 3rdterm is positive. Our 2groups are: (5x-10) + (x3-x2)2. Factor out the greatest commonfactor from each term:5(x-2)+(x2-1)3. Note that there is no commonbinomial factor. No groupingwill lead to a common factor –We cannot find a solution byfactoring by grouping.Example: 3xy+2-3x-2y1. Note that the first 2 terms haveno common factors other thanone. If we rearrange the terms,however, we can create 2groupings with commonfactors: (3xy-3x)+(-2y+2).NOTICE THAT WE HAVEA “-” IN FRONT OF THETHIRD TERM. Also note the“+” in front of the 4thterm. Wewill still factor out the ‘-’, sothat we do not lose track of it.This gives us: (3xy-3x)-(2y-2)2. Factor out the greatest commonfactor from each term: 3x(y-1)-2(y-1)3. Factor out a common binomialfactor. Our solution is:(y-1)(3x-2)Remember, factoring bygrouping can be efficient, but itdoesn’t work all that often. It is angood review of algebraicprocedures. BE CAREFUL whenthere is a “-” in front of the thirdterm. ALWAYS factor that out ofthe third and fourth terms whengrouping.Online ResourcesGCF & Factoring by Groupinghttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut27_gcf.htmInteract Tutorial: GCG &Factoring by Groupinghttp://www.mathnotes.com/intermediate/Mchapter05/aw_MInterAct5_7.htmlGCF & Factoring by GroupingGuide (.pdf file)http://online.math.uh.edu/Math1300/ch4/s41/ex41.pdfGCF & Factoring by Groupinghttp://online.math.uh.edu/Math1300/ch4/s41/GCF/Lesson/Lesson.htmlFactoring & Polynomialshttp://www.okc.cc.ok.us/maustin/Factoring/Factoring.htmlFactoring Strategieshttp://hhh.gavilan.cc.ca.us/ybutterworth/intermediate/ch5Angel.docShouldn’t your baby be a Gerbor baby?
- 12. Solving equations is anessential part of Algebra. Factoringequations breaks mathematicalstatements that look complex intosmaller parts. Often, what lookslarge, complex and difficultbecomes clear when we can look atit in smaller or simpler terms.A polynomial is a finite sum ofterms of the form axnwhere a is areal number and n is a whilenumber. Example: 5x3-6x2+3x-6A quadratic equation is anequation that can be written in theform ax2+bx+c=0 with a not equalto 0. When a quadratic equation iswritten in the form ax2+bx+c=0, itis called standard form.If we can write a polynomial inthis standard form, it is a quadraticequation; 2x2+5x = -3 is an exampleof a polynomial equation. Here, thesame equation is written in standardform:2x2+5x+3=0While the polynomial westarted with did not look like aquadratic equation, it is. Note thatthe degree of the above a quadraticequation is 2. This is a commontype of equation.Zero Factor PropertyIf ab = 0, then a = 0 or b = 0.Zero is the key, because the onlyway a product can become 0 is if atleast one of its factors is 0. Wewould not be able to make a generalstatement about the factors if theproduct was set equal to any othernumber. For example, if ab = 1,then a = 7 and b = 1/7 or a = 4 and b= 1/4, etc. But with the product setequal to 0, we can guarantee findingthe solution by setting each factorequal to 0. This is why standardform (ax2+bx+c=0) is so important.Solving a Quadratic Equations byFactoringStep 1: Write the equation instandard form. First, if necessary,simplify the equation (i.e. clear anyfractions or parenthesis () andcombine like terms BEFORErewriting in standard form)Step 2: Factor completelyStep 3: Set each factor containing avariable equal to 0.Step 4: Solve the resultingequations.Example: 3x2= 13x-4Step 1. Rewrite in standard form:3x2-13x+4 = 0Step 2. Factor completely:(3x-1)(x-4) = 0Step 3. Set each factor containing avariable equal to 0: 3x-1 = 0 andx+4 = 0Step 4. Solve the resultingequations:3x-1 = 0 x-4 = 03x = 0+1 x = 0+4x = 1/3 x = 4In this problem, there is nosimplifying – we were able toimmediately rewrite in standardform by using the addition propertyof equality.Example: x2= 121Step 1. Rewrite in standard form:x2-121 = 0. NOTE: This is adifference of squares;a2-b2= (a+b)(a-b)Step 2. Factor completely:(x-11)(x+11) = 0Step 3. Set each factor containing avariable equal to 0: x-11 = 0 andx+11 = 0Step 4. Solve the resultingequations:x-11 = 0 x+11=0x = 11 x = -11being able to recognizedifference of squares and perfectsquare trinomials is important.Example: x2+12x = -36Step 1. Rewrite in standard form:x2+12+36 = 0. NOTE: This is aperfect square trinomial with apositive middle term;a2+2ab+b2= (a+b)2Step 2. Factor completely:(x+6)2= 0Step 3. Set each factor containing avariable equal to 0: x+6 = 0. Note:In a perfect square trinomial, thereis only 1 factorStep 4. Solve the resultingequations:x + 6 = 0x = -6Mr. Breitsprecher’s Edition March 29, 2005 Web: www.clubtnt.org/my_algebraAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.
- 13. Because the first term andcoefficient were perfect squares andthe middle term was positive, if thistrinomial is factorable, we have aperfect square trinomial that is the sumof two terms.Example: 4m2-4m = -1Step 1. Rewrite in standard form:4m2-4m+1 = 0. NOTE: This is aperfect square trinomial with anegative term; a2-2ab+b2= (a-b)2Step 2. Factor completely: (2x-1)2= 0Step 3. Set each factor containing avariable equal to 0: 2x-1 = 0. Note:In a perfect square trinomial, there isonly 1 factor.Step 4. Solve the resulting equation:x-1 = 0x = 1/2Because the first term andcoefficient were perfect squares andthe middle term was negative, if thistrinomial is factorable, we have aperfect square trinomial that is thedifference of two terms.Mr. Breitsprecher’s Edition March 29, 2005 Algebra Connections, Page 2Example: 12x3+5x2= 2xStep 1. Rewrite in standard form:12x3+5x2-2x = 0.Step 2. Factor completely. Note:first, factor out the greatest commonfactor (GCF) x(12x2+5x-2). Thenfactor the trinomial:x(4x-1)(3x+2)=0Step 3. Set each factor containing avariable equal to 0: x = 0, 4x-1 = 0,and 3x+2 = 0.Step 4. Solve the resulting equations:x = 0 (4x-1) = 0 (3x+2) = 04x = 1 3x=-2x = 1/4 x = - 2/3Don’t forget your GCF! THIS ISALWAYS THE FIRST STEPWHEN FACTORING. If is containsa variable (or is a variable as in thiscase), be sure to set it equal to zerotoo.Example: x(4x-11) = 3Step 1. Rewrite in standard form:4x2-11x-3 = 0. NOTE: Before werewrite this polynomial in standardform, we must simplify the equation.Remove the parenthesis by distributingthe x through the factor (4x-11). Then,rewrite the resulting equation to beequal to 0.Step 2. Factor completely (4x+1)(x-3).NOTE: This is not a “perfect squaretrinomial” the first term cannot befactored as (2x)(2x).Step 3. Set each factor containing avariable equal to 0:4x+1 = 0 and x-3 = 0Step 4. Solve the resulting equations:4x+1 = 0 x-3 = 04x = -1 x = 3x = -1/4The Fundamental Theorem ofAlgebraLook at the examples given.Compare the number of solutions withthe degree of the polynomial. Thenumber of solutions to any polynomialequation is ALWAYS less than orequal to the degree of the polynomial.This fact is known as the fundamentaltheorem of algebra.Additional Key Concepts: Factoring PolynomialsThe Greatest Common FactorFactoring is the process of writing an expression as aproduct.The GCF of a list of common variables raised topowers is the variable raised to the smallest exponent inthe list.The GCF of a list of terms is the product of all commonfactors.Factoring by Grouping:1. Group the terms into two groups of two terms.2. Factor out the GCF from each group.3. If there is a common binomial, factor it out.4. If not, rearrange the terms and try steps 1-3 again.Factoring Trinomials in the Form x2+bx+cx2+bx+c = (x+?)(x+?), where the numbers indicated by the“?” sum to “b” and the product of the numbers indicatedby the “?” is “c”Factoring Trinomials in the Form ax2+bx+cTo factor ax2+bx+c, try various combinations of factors ofax2and c until the middle term of bx is obtained whenchecking (the product of the outside term and the productof the inside terms sum to equal the middle term).Factoring Trinomials in the Form ax2+bx+c by Grouping1. Find two numbers whose product is a*c and whosesum is b2. Rewrite bhx, using the factors found in step 1.3. Factor by grouping.Factoring Perfect Square Trinomials (Trinomials that arethe square of some binomial)a2+2ab+b2= (a+b)2a2-2ab+b2= (a-b)2Difference of Two Squaresa2-b2= (a+b)(a-b)Online ResourcesQuadratic Equationshttp://www.mathpower.com/tut99.htmQuadratic Equations: Solutions by Factoringhttp://www.sosmath.com/algebra/quadraticeq/sobyfactor/sobyfactor.htmlhttp://www.mathpower.com/tut105.htmhttp://www.mathpower.com/tut110.htm
- 14. Factoring is an idea you might befamiliar with from multiplication.Numbers that can be multiplied togetherto get another number are its factors. Forexample, 4*3 = 12, so 3 and 4 arefactors of 12. However, theyre not itsonly factors; 1, 2, 6, and 12 are otherfactors of 12. (Another way of defininga factor is a number that goes evenlyinto the number youre factoring.)A number is prime if it can not bedivided evenly by anything except itselfand 1. For example, 5 is a primenumber, because the only factors of 5are 1 * 5 = 5. However, 12 is not aprime number, because 1 * 12 = 12, 2 *6 = 12, and 3 * 4 = 12. Primefactorization means finding all the primenumbers that are factors of a number.Why is this important – it is thefoundation of what we will do the rest ofthe semester! We need to be able to findLowest Common Multiples (LCM). Inour “Review,” we saw that the LowestCommon Multiple is the LowestCommon Denominator (R-2).It will not be possible to work withfractions without this understanding.Please NEVER attempt to add orsubtract fractions unless they have acommon denominator!Suppose you want to find the LeastCommon Multiple (Please rememberLCM = LCD) of 6 and 10. If we start byfactoring them into primes, we get this:6: 2 * 310: 2 * 5Now the challenge is to find thesmallest possible set of prime factorsthat contains all the factors of eachoriginal number.In other words, we need a 2 and a 3from the 6. So far we have 2 * 3. Butwe dont need to include another 2 fromthe 10 since we already have the 2 fromthe 6.All we need to add to our set is the5, so now we have 2 * 3 * 5. As acheck, can you find 2 * 3 in our set of 2* 3* 5? How about 2 * 5? Yes, bothoriginal factor sets are included andthere is nothing extra. So, our LCM is 2* 3 * 5 or 30.Lets try another one. Suppose youare looking for the LCM of 8, 10 and 12.Start by factoring each one out:8: 2 * 2 * 210: 2 * 512: 2 * 2 * 3Now lets put together the smallestset that contains each of those three sets.We start with three 2s from the 8. Thatgives us: 2 * 2 * 2For the 10, we certainly dont needanother 2 since we already have three ofthem, but we do need to add in a 5.Now we have: 2 * 2 * 2 * 5For the 12, we dont need to doanything with the two 2s because wealready have three of them, but we doneed to toss in the 3. Now we have: 2 *2 * 2 * 5 * 3Again, can you find each of thethree smaller factor sets within thatbigger one? Yes, so were in business.The LCM is 2 * 2 * 2 * 5 * 3 or 120.Source: mathforum.orgBeing able to think about numbersis the first step towards being able towork with them. Fundamental to numbertheory are numbers themselves, and thebasic building blocks for numbers areprime numbers.A prime number is a countingnumber that only has two factors, itselfand one. Counting numbers which havemore than two factors (such as six,whose factors are 1, 2, 3 and 6), are saidto be composite numbers.The number one only has onefactor and is considered to be neitherprime nor composite.You can learn EVERYTHING youcould ever imagine about primenumbers at:http://www.utm.edu/research/primes/Here are some prime numbers startingwith 2:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83,89, 97, 101, 103, 107, 109, 113, 127,131, 137, 139, 149, 151, 157, 163, 167,173, 179, 181, 191, 193, 197, 199, 211In this class, you will benefit if youcan recognize prime numbers and factorsup to at least 59.MS EXCEL and Primes?Can you generate these lists from MSExcel? Well, no – not without special“plug-ins.” Software to generate primes isavailable FREE as shareware.Prime Derivatives is one suchprogram. It is available at:http://www.sharewareconnection.com/prime-derivatives.htmThis attractive, handy little primenumber calculator takes any number up totwenty digits, and then factorizes all theunique prime numbers that evenly divideinto the number that was entered into thefield. Its small enough to neatly fit in thecorner of your monitor while you easilyfactorize the numbers you need for thatschool or office assignment.A great interactive site to check primefactorizations is at:http://www.gomath.com/algebra/factor.phpMr. Breitsprecher’s Edition January 20, 2005 Web: www.clubtnt.org/my_algebra
- 15. Factoring a polynomial is theopposite process of multiplyingpolynomials. Factor means write asa product. Recall that when wefactor a number, we are looking forprime factors that multiply togetherto give the number; for example6 = 2 * 3, or 12 = 2 * 2 * 3.When we factor a polynomial,we are looking for simplerpolynomials that can be multipliedtogether to give us the polynomialthat we started with. Understandinghow to multiply polynomials is thekey to understanding how to factorthem.When we factor a polynomial,we are usually only interested inbreaking it down into polynomialsthat have integer coefficients andconstants.Simplest Case: RemovingCommon FactorsThe simplest type of factoring iswhen there is a factor common toevery term. In that case, you canfactor out that common factor. Whatyou are doing is using thedistributive property in reverse.Recall that the distributive law saysa(b + c) = ab + ac.Source: www.jamesbrennan.org/algebraThinking about it in reverse meansthat if you see ab + ac, you can writeit as a(b + c).Example: 2x2+ 4xNotice that each term has a factor of2x, so we can rewrite it as:2x2+ 4x = 2x(x + 2)Difference of Two SquaresIf you see something of the form(a2- b2), you should remember theformula(a+b)(a-b) = a2-b2Example: x2– 4 = (x – 2)(x + 2)This only holds for aDIFFERENCE of two squares.There is no way to factor a sum oftwo squares such as a2+ b2intofactors with real numbers. Tryingto do so is a common mistake inalgebra – please avoid this one!Trinomials (Quadratic)A quadratic trinomial has the formax2+ bx + c,where the coefficients a, b, and c,are real numbers (for simplicity wewill only use integers, but in real lifethey could be any real number). Weare interested here in factoringquadratic trinomials with integercoefficients into factors that haveinteger coefficients.Not all such quadraticpolynomials can be factored usingthe real numbers, and even fewerinto integers. Therefore, when wesay a quadratic can be factored, wemean that we can write the factorswith only integer coefficients.If a quadratic can be factored,it will be the product of two first-degree binomials, except for verysimple cases that just involvemonomials. For example x2byitself is a quadratic expressionwhere the coefficient a is equal to 1,and b and c are zero. We know thatx2factors into (x)(x).Another situation occurs whenonly the coefficient c is zero. Thenyou get something that looks like2x2+ 3xThis can be factored verysimply by factoring out(‘undistributing’) the commonfactor of x:2x2+ 3x = x(2x + 3)The most general case is whenall three terms are present, as inx2+ 5x + 6Mr. Breitsprecher’s Edition March 3, 2005 Web: www.clubtnt.org/my_algebraAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.
- 16. We look at two cases of this type.The easiest to factor are the oneswhere the coefficient of x2(which weare calling ‘a’) is equal to 1, as in theabove example. This is the simplestcase; let’s begin by looking at a = 1examples.Coefficient of x2is 1Since the trinomial comes frommultiplying two first-degreebinomials, let’s review what happenswhen we multiply binomials using theFOIL method. Remember that to dofactoring we will have to think aboutthis process in reverse (you could saywe want to ‘de-FOIL’ the trinomial).Suppose we are given(x + 2)(x + 3)Using the FOIL method, we get(x + 2)(x + 3) = x2 + 3x + 2x + 6Then, collecting like terms gives(x + 2)(x + 3) = x2 + 5x + 6Notice where the terms in thetrinomial came from. The x2camefrom x times x. The interesting part iswhat happens with the other parts, the‘+ 2’ and the ‘+ 3’.The last term in the trinomial, the6 in this case, came from multiplyingthe 2 and the 3. Where did the 5x inthe middle come from? We got the 5xby adding the 2x and the 3x when wecollected like terms. We can state thisas a rule:If the coefficient of x2is one, then tofactor the quadratic you need to findtwo numbers that:1. Multiply to give the constant term(which we call c)2. Add to give the coefficient of x(which we call b)This rule works even if there areminus signs in the quadraticexpression, if we remember how toadd/subtract and multiply/dividepositive and negative numbers.Special Case: Perfect SquareTrinomialRecall from special products ofbinomials :(a+b)2= a2+2ab+b2(a-b)2= a2– 2ab + b2The trinomials on the right are calledperfect squares because they are theMr. Breitsprecher’s Edition March 3, 2005 Algebra Connections, Page 2squares of a single binomial, ratherthan the product of two differentbinomials. A quadratic trinomial canalso have this form:(x + 3)2 = (x + 3)(x + 3) = x2+ 6x + 9Notice that the coefficient of x is thesum 3 + 3, and the constant term is theproduct 3 * 3. One can also say:The coefficient of x is twice thenumber 3The constant term is the numberthree squaredIn general, if a quadratic trinomial is aperfect square, thenThe coefficient of x is twice thesquare root of the constant termOr to restate it another way,The constant term is the square ofhalf the coefficient of xIn symbolic form, we can express thisas:(x+a)2= x2+2ax+a2It is important to be able to recognizeperfect square trinomials. This is thekey to solving quadratic equations.Coefficient of x2is not 1A quadratic is more difficult to factorwhen the coefficient of the squaredterm is not 1, because that coefficientis mixed in with the other productsfrom FOILing the two binomials.There are two methods for attackingthese: either you can use a systematicguess-and-check method, or a methodcalled factoring by grouping. We willfirst look at the guess-and-checkmethod.If you need to factor a trinomial suchas2x2+ x - 3Think about what combinationscould give the 2x2as well as the othertwo terms.In this example the 2x2mustcome from (x)(2x), and the constantterm might come from either (-1)(3)or (1)(-3). Now we must figure outwhich combination will give thecorrect middle term. By “trial anderror” we get:(x-1)(2x+3)Notice these patterns.The first term in the trinomial(the 2x2) is just the product of theleading terms in the binomials.The constant term in the trinomial(the -3) is the product of theconstant terms in the binomials(so far this is the same as in thecase where the coefficient of x2is1)The middle term in the trinomial(the x) is the sum of the outer andinner products. This involves allthe constants and coefficients andis not always obvious.Because 1 and 2 are relatively simpleand 3 is complicated, it makes sense tothink of the possible candidates thatwould satisfy conditions 1 and 2, andthen test them in every possiblecombination by multiplying theresulting binomials to see if you getthe correct middle term.Step-by-Step Process1. List all the possible ways to getthe coefficient of x2(which wecall “a”) by multiplying twonumbers2. List all the possible ways to getthe constant term (which we call“c”) by multiplying two numbers3. Try all possible combinations ofthese to see which ones give thecorrect middle termDon’t forget that the numberitself times 1 is a possibilityIf the number (a or c) isnegative, remember to try theplus and minus signs bothwaysIn our example: 2x2+ x – 3, we makea list of the possible factors of 2x2:The only choice is (2x)(x).Then we make a list of the possiblefactors of the constant term -3: it iseither (1)(-3) or (-1)(3).The possible factors of the trinomialare the binomials that we can make outof these possible factors, taken inevery possible order.(2x + 1)(x – 3)(x + 1)(2x – 3)(2x + 3)(x – 1)(x + 3)(2x – 1)Multiplying these and find the one thatworks (the third one). All you reallyneed to check is to see if the sum ofthe outer and inner multiplicationswill give you the correct middleterm, since we already know that wewill get the correct first and last terms.
- 17. Let’s look at factoringtrinomials with a “leadingcoefficient other than 1” We canwrite that form as ax2+ bx + c,where a, b, and c are integers.Like factoring any otherpolynomial, the first thing to do is tofactor out all constants which evenlydivide all three terms (GreatestCommon Factor). If “a” is negative,factor out -1.This will leave an expression inthe form: d(ax2+ bx + c), where a,b, c, and d are integers, and a > 0.Now we have a factor (GCF) times atrinomial in the form ax2+ bx + c.The next step is to factor theremaining trinomial. We willpresent 2 methods.The first method, which will bereferred to as “Old SchoolAlgebra.” NOTE THAT THIS ISNOT THE METHOD THAT ISPRESENTED IN OURTEXTBOOK. It is similar to thethe method presented in ourtextbook (unit 4.3), but uses adifferent set of “tests” utilizingabsolute values when “c” is positiveand when “c” is negative.It is presented here as analternative perspective – studentsneed not study this method. Likethe method presented in our text,unit 4.3, it comes down to “trial-by-error.”Our text presents a second,similar method, which I call “Guessand By Golly” (Unit 4.3). Like the“Old School” method, it involvesdetermining possible factors andchecking them until one works.Our text’s method differs in how itlooks at positive and negative valuesfor “c.” Rather than creating a set ofrules about when “c” is positive andwhen “c” is negative, “Guess and ByGolly” relies on an understanding ofpatterns with signs.The third method, “Factoringby Grouping,” (Unit 4.4 in our text)works by rewriting our trinomialinto a 4 term polynomial that can befactored by groups.Old School AlgebraHere is how to factor anexpression ax2+ bx + c, where a > 0(“a” is positive):1. Write out all the pairs ofnumbers that, when multiplied,produce “a” (i.e. a1*a2=a,a3*a4=a, etc.).2. Write out all the pairs ofnumbers that, when multiplied,produce “c” (i.e. c1*c2=c,c3*c4=c, etc.).3. Pick one of the a pairs, (a1, a2),and one of the c pairs, (c1, c2).4. If c > 0 (“c” is positive):Compute a1c1 + a2c2. If | a1c1 +a2c2 | = b, then the factored formof the quadratic is:(a1x + c2)(a2x + c1) if b > 0(“b” is positive).(a1x - c2)(a2x - c1) if b < 0(“b” is negative).5. If a1c1 + a2c2 ≠ b, compute a1c2+ a2c1. If a1c2 + a2c1 = b, thenthe factored form of thequadratic is (a1x + c1)(a2x + c2)or (a1x + c1)(a2x + c2). If a1c2 +a2c1 ≠ b, pick another set ofpairs.6. If c < 0 (is negative). Computea1c1 -a2c2. If | a1c1 - a2c2 | = b,then the factored form of thequadratic is (a1x - c2)(a2x + c1)where a1c1 > a2c2 if b > 0 (ispositive) and a1c1 < a2c2 if b <0 (is negative).7. Using FOIL, the outside pairplus (or minus) the inside pairmust equal b. This is a check.Example 1: 3x2- 8x + 4Numbers that produce 3: (1, 3).Numbers that produce 4: (1, 4),(2, 2).Continuing:(1, 3) and (1, 4): 1(1) + 3(4) =11≠8. 1(4) + 3(1) = 7≠ 8.(1, 3) and (2, 2): 1(2) + 3(2) = 8.(x - 2)(3x - 2).Check: (x - 2)(3x - 2) = 3x2 -2x -6x + 4 = 3x2 - 8x + 4.Example 2: Factor 12x2+ 17x + 6Numbers that produce 12: (1, 12),(2, 6), (3, 4).Numbers that produce 6: (1, 6),(2, 3).Continuing:(1,12) and (1,6): 1(1)+12(6) = 72.Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Mr. Breitsprecher’s Edition March 16, 2005 Web: www.clubtnt.org/my_algebra3Methods,YouChoose!
- 18. Mr. Breitsprecher’s Edition March 16, 2005 Algebra Connections, Page 21(6) + 12(1) = 18.(1, 12) and (2, 3): 1(2) + 12(3) =38. 1(3) + 12(2) = 27.(2, 6) and (1, 6): 2(1) + 6(6) = 38.2(6) + 6(1) = 18.(2, 6) and (2, 3): 2(2) + 6(3) = 22.2(3) + 6(2) = 18.(3, 4) and (1, 6): 3(1) + 4(6) = 27.3(6) + 4(1) = 22.(3, 4) and (2, 3): 3(2) + 4(3) = 18.3(3) + 4(2) = 17.(3x + 2)(4x + 3).Check: (3x + 2)(4x + 3) = 12x2+9x + 8x + 6 = 12x2+ 17x + 6.Example 3: Factor 4x2- 5x - 21Numbers that produce 4: (1, 4),(2, 2).Numbers that produce 21: (1, 21),(3, 7).Continuing:(1,4) and (1,21): 1(1)-4(21)= - 83.1(21) - 4(1) = 17.(1,4) and (3,7): 1(3) - 4(7) = - 25.1(7) - 4(3) = - 5.(x - 3)(4x + 7).Check: (x - 3)(4x + 7) = 4x2+7x -12x - 21 = 4x2- 5x - 21.Analysis: “Old School”Probably, most will find thisprocedure complex – it requires a adifferent rule depending on the signof “c.” Mr. B’s AlgebraConnections DOES NOTRECOMMEND STUDENTS USETHIS METHOD!It is actually the same processpresented in our text, but it relies ona different set of rules usingabsolute values) depending on thesign of “c.” Many will finddifferent set of rules (depending onthe sign of “c”) frustrating – itseems like more to memorize.For most of us, fewer rules arebetter. Our text uses the sameconcepts, but relies on anunderstanding of sign patterns whenfactoring.The key is to remember: anegative number MUST havefactors with different signs (onepositive, the other negative). Apositive number MUST have factorswith the same signs (either bothpoitive or both negative) Seeingpatterns in numbers is important inalgebra. It is often easier to learnthese if we allow someone to“guide” our practice.Factor by Grouping (Unit 4.4)1. Factor out the GCF, if it exists.This should always be your firststep in EVERY factoringproblem (no matter whatmethod you choose to use).2. Find 2 "mystery" numbers.These are similar to thenumbers you sought whenfactoring simpler trinomials -but there is a slight difference.You still want the sum to be thex-coefficient (b), but now youwant their product to equal ac,the product of the leadingcoefficient and the constant.3. Replace the x-coefficient.Rewrite the polynomial, butwhere b once stood, write thesum of the two "mystery"numbers in parentheses. This isthe key - we will now "blowthis trinomial up" into twobinomials.4. Distribute x through theparentheses. Theres still an xmultiplied times the quantity inparentheses from step 3multiply each of the mysterynumbers by x to eliminate thoseparentheses. AT THIS POINTWE HAVE A FACTOR BYGROUPING PROBLEM.5. Factor by Grouping.Example: 6x2-x-12Solution: This polynomial has noGCF. Skip right to calculating the"mystery" numbers; they should addto “b,” in this case, -1 and have aproduct of 6 * (-12) = -72. Thosenumbers, therefore, must be -9 and8. Replace the x-coefficient of -1with the sum of those "mystery"numbers in parentheses (groupingsymbols). This leaves us with:6x2+(-9+8)x-12Distribute the x lying outsidethe parenthesis: 6x2-9x+8x-12Now you can factor by grouping:= 3x(2x-3) + (3x+4)= (2x-3)(3x+4)Analysis: Factor by GroupingThis procedure is direct andappears simpler than other methods.It is based on re-writing trinomialsinto 4 term polynomials (theopposite of simplifying) and issometimes called “decomposition.”There is not as much "guess andby golly" or "trial and error" that isused in the textbook’s method or the“old school” approach. Be carefulabout deciding whether The Bombor our textbooks method is simpler."Trial and error" might seem likeextra work, but with practice, youwill see patterns and choose theright solution more and more often.Our textbook shows exampleswhere we work out every possiblesolution WRONG before we findthe right one. As we learn the“patterns,” we will tend to start withthe RIGHT solution withoutlooking at all of the wrong ones.THIS IS WHY GUIDEDPRACTICE, working with anotherperson (like your professor or theMath Center), is important.Please remember, text usescarefully selected problems toillustrate important patterns innumbers - patterns that reinforcealgebra skills and concepts. Ourtextbook is a beautiful presentationof a complex subject. Pleasepractice Units 4.3 and 4.4. Thiswill build your algebra skills andthen you can decide which to use.To factor ax2+bx+c, try various combinations of factors of ax2and c until a middle term of bx is obtained when checking. K.Elayn Martin-Gay has written our textbook to illustrate these“patterns.” Please practice Unit 4.3 with the Math Center or me.Working together, this presentation will work for you too!Example: 3x2+ 14x - 5Factors of 3x2: (3x, x); Factors of –5: (-1,5) & (1,-5)(3x-1)(x+5) checks – the product of the outer terms (15x)plus the product of the inner terms (-1x) equals the middleterm of our trinomial (14x). Our solution is (3x-1)(x+5)
- 19. Factoring Trinomials in theform, where “b” and “c” are integersis a good place to start factoringtrinomials. Notice that the firstcoefficient (x`) is 1. Let’s look atthe patterns and procedures we willneed to solve these problems. Thiswill help us when we will looktrinomials where the leadingcoefficient is other than 1.When we factor trinomials witha lead coefficient of 1, we will endup with a factorization in the formof 2 binomials: (x+?)(x+?). If wecan determine the numbers thatreplace the “?” in each binomial, wehave factored the trinomial.Remember, IF THETRINOMIAL IS FACTORABLE(x2+bx+c), there are 2 numberswho’s SUM is “b” and who’sPRODUCT is “c” Just look for apair of numbers that will sum to “b”and then check their product to seeif it is “c.” This should determine ifthe trinomial is factorable and, if so,will fill in the missing terms of our 2binomials: (x+?)(x+?).Alternatively, find 2 numberswho’s product is “c” and then checktheir sum to see if it is “b.” Eitherway, once we see these “patterns,”we can use them to help identify themissing terms when we factor atrinomial in the form of of x2+bx+cinto (x+?)(x=?). We will focus onthis alternative perspective.The following steps can be usedto factor trinomials in the formx2+bx+c:1. If possible, factor out anygreatest common factor (otherthan 1 or –1)2. Look at each possible sets of 2factors that will result in “c”3. Add each of those 2 factors andidentify which set sums to “b”4. Write our binomial factors byfilling in the missing terms:(x+?)(x+?)Patterns in the SignsIn many ways, algebra is allabout seeing patterns in numbers.When factoring trinomials in theform of x2+bx+c, look at the signsof “b” and “c” and notice:• If “c” is positive, then thenumbers we need to complete ourfactorization [(x=?)(x+?)] havethe same sign – they are bothpositive or negative. We see:If “b” is positive, then bothfactors of “c” must be positive.If “b” is negative, then bothfactors of “c” must be negative.• If “c” is negative, then one of thenumbers we need to complete ourfactorization [(x=?)(x+?)] must bepositive and the other must benegative (two opposite signs).Factor Out GCF FIRST!Don’t forget, wheneverfactoring a polynomial – ALWAYSFACTOR OUT ANYGREATEST COMMONFACTOR before attempting anyother factorization.Example: x2+6x+8Note that “c” is positive (8).This tells us (if the trinomial isfactorable) that the numbers neededto complete our factorization[(x+?)(x+?)] must have the samesign. Because “b” is positive, weknow that we are looking at twofactors of “c” (or 8) that have to bepositive.We now know our choices are(2*4) and (1*8). We need to lookat which of these factors (if any)will sum to “b,” in this case, 6.Note that (2+4=6) and (1+8=9).We now know that this trinomial isfactorable and our solution is(x+2)(x+4).Example: x2-11x+10Again, “c” is positive (10).This tells us (if the trinomial isfactorable) that the numbers neededto complete our factorization[(x+?)(x+?)] must have the samesign. This time, because “b” isAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Mr. Breitsprecher’s Edition March 15, 2005 Web: www.clubtnt.org/my_algebra
- 20. Mr. Breitsprecher’s Edition March 15, 2005 Algebra Connections, Page 2negative (-11), we know that we arelooking at two factors of “c” (or 10)that have to be negative.We now know our choices are(-2*-5) or (-1*-10). Now we needto look at which of these factors (ifany) will sum to “b,” in this case, -11. Note that (-2 + -5 = -7) and (-1+ -10 = -11). We now know thatthis trinomial is factorable and oursolution is (x + -1)(x + -10). Wewould rewrite this. Our solution is(x-1)(x-10)Example: x2-3x-10Here we see that “c” is negative(-10). This tells us (if the trinomialis factorable) that the numbersneeded to complete our factorization[(x+?)(x+?)] must have different oropposite signs (one positive, theother negative). We now know thatour choices are (-1 * 10), ( 1*- 10),(-2*-5), or (-2*5). Now we look ateach of these factors to see which (ifany) will sum to “b,” in this case –3.The only pair of these factors thatwill sum to –3 is (2+ -5). Therefore,this trinomial is factorable as(x+2)(x+ -5). We would rewritethis. Our solution is (x+2)(x-5).Example: 2x3-24x2+64xNote that this one, at firstglance, does not look like atrinomial in the form x2+bx+c; but itis! Remember to always look for agreatest common factor BEFOREattempting any other factorization ofa polynomial. In this case, we havea GCF of 2x. We would start byrewriting this polynomial as 2x(x2-12x+32). Now we can factor it likethe other examples.Note that “c” is positive (32).This tells us (if the trinomial isfactorable) that the numbers neededto complete our factorization[(x+?)(x+?)] must have the samesign. Because “b” is negative (-12),we know that we are looking at twofactors of “c” (or 32) that have to benegative.Our choices are (-1 * -32),(-2* -16) or (-4 * -8). Only the lastof these three choices will sum toour “b” term (-4 + -8 = -12)Therefore, this trinomial isfactorable (DON’T FORGET OURGCF) as 2x(x+ -4)(x+ -8). Wewould rewrite this. Our solutionis 2x(x-4)(x-8).Example: 4x2+36x+80Again, at first glance, this doesnot look like a trinomial in the formx2+bx+c; but it is! Remember toalways look for a greatest commonfactor BEFORE attempting anyother factorization of a polynomial.In this case, we have a GCF of 4.We would start by rewriting thispolynomial as 4(x2+9x+20). Nowwe can factor it like the otherexamples.Note that “c” is positive (20).This tells us (if the trinomial isfactorable) that the numbers neededto complete our factorization[(x+?)(x+?)] must have the samesign. Because “b” is positive (9),we know that we are looking at 2factors of “c” (or 20) that have to bepositive.We know our choices are(1*20), (2*10) or (4*5). Only thelast of these three choices will sumto our “b” term (4+5=9) Therefore,this trinomial is factorable (don’tforget our GCF) and our solution is4(x+4)(x+5).Not Just “x”Just because we illustrate theform of a trinomial with a degree of2 and “leading coefficient of 1” as:x2+bx+c, any variable can bereplace the “x,” (i.e. “w” or “z”) justas any number can be the coefficientof our 2ndterm (we called it “b”) andany number can be our constant (wecalled it “c”).Online ResourcesIn most cases, you can go to the“domain” (i.e. http://domainname.ext)and follow links to the pagesidentified. ASK ME TO SENDTHEM TO YOU VIA EMAIL ASLINKS IF YOU HAVE ANYPROBLEMS!Factoring Power Pointshttp://students.loyola.ca/classes/powellt/Math4_5/factring.ppthttp://www.tcc.edu/vml/Mth03/Trinom/documents/FactoringTrinomials.ppsSimple Trinomials as Product ofBinomials (Many examples, printable)http://www.math.bcit.ca/competency_testing/testinfo/testsyll11/basicalg/basops/factoring/trinom/trinom.docFactoring Trinomials (Somepresentations do not treat trinomialsin the form of x2+bx+c, or leadingcoefficient of 1, as different fromax2+bx+c. We have started with thesimplest form, before introducingother trinomials.http://www.mathmax.com/introalg/chapter/bk3c5im.htmlhttp://www.coolmathalgebra.com/Algebra1/10FactDivPolys/04_undoingFOIL.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut28_facttri.htmhttp://www.jamesbrennan.org/algebra/polynomials/factoring_polynomials.htmOnline Factoring Solutions (Enterexponents as x^2)http://www.webmath.com/factortri.htmlhttp://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=algebra&s2=factor&s3=basic (Note: Domain forthis one is http://www.quickmath.com/)Factoring. Process of writing an expression as a product.Greatest Common Factor. Product of ALL commonfactors from a set of terms. When there are commonvariable factors in terms, use the common variable(s) raisedto the smallest exponent(s) of the in the terms.Special Products(a+b)2= a2+ 2ab + b2(a-b)2= a2– 2ab + b2(a-b)(a+b) = a2– b2Common Errors(a+b)2≠ a2+ b2(a-b)2≠ a2– b2We forgot middle terms!
- 21. Before working with formulas,let’s look at some units ofmeasurement – many formulas willbe based on units. In the US, we usefeet, yards, and miles.12 inches = 1 foot; 3 feet = 1yard; 5,280 feet = 1 mile. Unitsused in formulas must be the same –so changing from one unit intoanother equivalency is important.Here are the conversions:Feet to Inches. Number offeet * 12Inches to Feet. Number ofinches/12Yards to Feet. Number ofyards * 3Miles to Feet. Numbermiles * 5,280Metrics are used throughout theworld and are universally acceptedin the sciences. These measures areall based on a system of 10’s – wewill look at them in another editionof Mr. Breitsprecher’s AlgebraConnections.Often, proportions are the bestway to solve measurementproblems. Recall that a proportionis a statement that 2 ratios are equal.The key is to use the same units ineach ratio, identify 3 of the figuresin the proportion, and solve for thefourth. The cross product is animportant shortcut to help us set upthese problems (see previous editionMr. Breitsprecher’s AlgebraConnections for more on this)Once we are comfortableworking with units and performingconversions, we are ready to workwith formulas. There is nothingnew here – identify the knownquantities and solve for theunknown using the algebraicprocedures reviewed in class. Hereare some common formulas that willbe used in many practical situationsand in Algebra.PerimeterThe distance around the outsideof a given area, or 2 dimensionalshape, is called the perimeter.Think of it as the total length of theborder around that shape. Here aresome common perimeter formulas.Perimeter of a Triangle. Aclosed figure with three straightsides is called a triangle. We canrefer to each side as side 1, side 2,and side 3. The perimeter is the sumof these sides. The formula is:P = s1+s2+s3Perimeter of a Square. A 4-sided, closed figure with each cornermeasuring 90 degrees (right angle)is a square. While we could expressthe formula for the perimeter as thesum of the 4 sides, this would lookawkward – it is easier to express itin terms of multiplication. Theformula is: P = 4sPerimeter of a Rectangle. A4-sided, closed figure with eachcorner measuring 90 degrees (rightangle) is a rectangle. We could alsoexpress its perimeter as the sum ofthe sides, but that would also lookawkward – it is easier to express itin terms of multiplication. Theformula is: P = 2l (length)+2w(width).Perimeter of a Circle. Acircle has a perimeter, but we call itthe circumference. This wordderives from circumstance – thinkof the situations around you. Thereare no sides to measure, so we usethe diameter (D) – a line drawnthrough the center point of a circlethat touches both sides of thefigure. Sometimes, we are onlygiven the radius (r), which is halfthe diameter (D/2). We need aspecial mathematical figure for thisformula, π. For most purposes, wecan approximate this with 3.14 –though it really will go on and onforever without repeating. Whenconvenient, we can also use thefraction 22/7. The formula for thecircumference of a circle is: C=πDAreaA measurement of how many2-dimensional units (squares) aparticular object or surface coversis called the area. Think of it as theflat space a shape occupies. It ismeasured in square units, usuallysquare inches, square centimeters,square feet, or square miles, and soforth. Imagine a floor covered withsquare tiles that are each 1-foot x 1foot. If we count the tiles, we havethe area in square feet.Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Mr. Breitsprecher’s Edition October 3, 2005 Web: www.clubtnt.org/my_algebra
- 22. Mr. Breitsprecher’s Edition October 3, 2005 Algebra Connections, Page 2Area of a Triangle. Think ofthe area of a triangle as ½ that of animaginary rectangle that includesthe triangle. We need to know thebase (b) or bottom of the triangle.We also need the height (h), theperpendicular distance from the baseto the top angle of the triangle. Theformula is: A = 1/2 bhArea of a Square. Because oftheir square corners (90 degrees)and opposites sides, the area of asquare is the same formula as thearea of a rectangle. It is theproducts of 2 sides (remember, theyare equal). The formula is:A = l (length) * w (width)Area of a Rectangle. For anexplanation, please see above. Theformula is: A = l (length) * w(width).Area of a Circle. The area of acircle is based on the radius (r),which is ½ the diameter (D/2). Theformula is A=πr2VolumeArea is a 2 dimensional, or flatmeasurement. Volume is a 3-dimensional measurement. Think ofit as being based on themeasurement across, front-to-back,and up-and-down. Imaginecounting how many 1-inch squaresugar cubes would fit in a box – thatis what volume does. Volume isalways measure in cubed units(cubic inches, cubic feet, cubiccentimeters, and so forth).Volume of Pyramid (ThreeDimensional Triangle). Thevolume of a pyramid with a squarebase is one-third the area of the base(b) multiplied by the height (h). Theformula is: V= 1/3(s2)*hVolume of a Square/Cube.The 90-degree corners and equalsides make this one straightforward.Note that a 3-dimensional square iscalled a cube. The formula is: V = s3Volume of a Rectangle. Thisone is based on the same idea as thevolume of a square – it’s the productof the length, width, and height.The formula is:V=l(length) *w(width)*h(height)Volume of a Sphere. Thisexpression, like the area, is based onthe radius (r), which is ½ thediameter (D). We also need to usepi. The formula is: V = 4/3(πr3)Volume of a Cylinder. Thinkof this shape as a 3-dimensionalcross between a rectangular box anda sphere. The volume is based onthe radius (r), the height (h) and pi.The formula is: V = πr2hVolume of a Cone. This shapeis a 3-dimensional cross between atriangle and a circle and is based onthe radius (r), the height (h) and pi.The formula is: V = 1/3(πr2h)Temperature ConversionsBoth Fahrenheit and Celsius arebased on the freezing and boilingpoints of water. Fahrenheit uses 32for freezing and 212 for boiling.Celsius uses 0 for freezing and 100for boiling. It is probably easiest towork with Celsius, but we typicallyuse Fahrenheit.Fahrenheit to Celsius.Subtract 32 from the F temperature,multiply by 5 and divide by 9. Theformula is: C= (5/9) (F-32)Celsius to Fahrenheit.Multiple the Celsius temperaturesby 9, divide by 5, and add 32. Theformula is: F = (9/5)C+32More Useful FormulasDistance. The formula forcalculating distances is based on rate(speed expressed as a ratio ofdistance per unit of time) and time.Be sure that the distance isexpressed in the same unit as therate. Read these problems carefully– identify the variables that areknow, express all variables based onthe same units, and then plug theknown variables (with the sameunits) into the formula and solve forthe unknown. The formula is:d = r (rate)*t (time)Percent. Always representinga ratio of some percentage to 100,we calculate this based on the baseand rate. If we express ratios interms of 100, we make comparisonsmore meaningful. When 2 ratios areinvolved, solve percentage problemswith proportions using the crossproduct. The formula is:p (percentage) = b (base) * r(rate)Pythagorean Theorem. Aspecial case of a triangle is when 1angle equals 90 degrees – this is acalled right triangle. ThePythagorean Theorem states that ina right triangle, the sum of thesquares of the lengths of the legs isequal to the square of the length ofthe hypotenuse (the side directlyopposite of the 90 degree or rightangle). Many distance and heightproblems can be solved by applyingthis theorem. The formula isc2(hypotenuse) = a2(side adjacentright angle) + b2(other sideadjacent right angle)Simple Interest. We expect topay people for the privilege ofborrowing money – we pay backmore than we borrow. The “extra”amount is “interest.” There aredifferent ways to calculate this;some are complex. Simple interestis computed only on the originalamount of a loan. To calculate thisamount, we need to know how muchis borrowed (principal), the annualinterest rate (expressed as a decimal,NOT as a percent), and the amountof time the money will be borrowedin years (if given in months, divideby 12 to express as part of a year).The formula is:I (interest) = P (principal)*r(annual rate) * t (time in years)Perimeter• Triangle: P = s1+s2+s3• Square: P = 4s• Rectangle: P = 2l + 2w• Circle: C=πDArea• Triangle: A = 1/2 bhSquare: A = lw• Rectangle: A = lw• Circle: A=πr2Volume:• Pyramid (4--sided base) = 1/3(s2)*h• Square/Cube = s3• Rectangle = lwh• Sphere = 4/3(πr3)• Cylinder = πr2h• Cone = 1/3(πr2h)Temperature Conversions• Fahrenheit to Celsius, C= (5/9) (F-32)• Celsius to Fahrenheit, F = (9/5)C-32Distance• d = rtPercent• p = brSimple Interest• I = PrtNote: Many other geometric shapes can be considered combinationsof these. When appropriate, use these formulas for each part.
- 23. A fraction has 2 main parts, thenumerator (top) and denominator(bottom). The line between the twois called the “fraction bar.” Need alittle “trick” to keep this straight?Just think “N” for North (up),numerator; “D” for Down,denominator.DN=43The denominator (down) orlower part of a fraction indicateshow many equal portions or piecesof the whole there are. Note thatwhen we divide a pizza into ½, eachpiece or half is of equal size.The same is true if we cut achunk of fudge into 4 pieces or ¼slices. Each piece of fudge will beequal. The numerator (north or up)refers to how many of those equalpieces the fraction represents.Types of FractionsWhen we start dividing wholethings into pieces, we have threesituations: proper, improper, andmixed fractions. Let’s review each.Proper Fractions. Thenumerator is always smaller than thedenominator. Most of us think ofthis type of fraction when thinkingof parts of a whole. This type offraction always represents less than1. This is the simplest type offraction to work with.Examples: Proper Fractions17612587ororImproper Fractions. Thenumerator is always larger than thedenominator. This fraction seems“top-heavy” or an odd way toexpress parts of a whole, becausethe fraction is actually greater than1. The denominator (down) stilltells us what size the equal partsare. The numerator (north or up)tells us that we have more thanenough pieces to make a whole –this fraction always representssomething greater than 1. This typeof fraction can be useful to workwith in many situations.Examples: Improper Fractions72435712ororMixed Fractions. Looking atimproper fractions can beconfusing, that’s why we havemixed fractions. They clearlyindicate how many parts of thewhole there are with proper fractionand use a whole number to indicatehow many complete “wholes” are.This is much easier for most of usto interpret than when looking atMr. Breitsprecher’s Edition March 3, 2005 Web: www.clubtnt.org/my_algebraAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.½¼Lowest Common Multiple = Lowest Common DenominatorThe Lowest Common Multiple (LCM) is the smallest number that is a commonmultiple of two or more numbers. When working with fractions, this is also theLowest Common Denominator (LCD). For example, the L.C.M of 3 and 5 is 15. Thesimple method of finding the L.C.M of smaller numbers is to write down the multiplesof the larger number until one of them is also a multiple of the smaller number.Example: Find the Lowest Common Multiple of 8 and 12.Solution: Multiples of 12 are 12, 24... 24 is also a multiple of 8, so the L.C.M of 8and 12 is 24.Example: LCM/LCD of Big Numbers. Find all the prime factors of both numbers.Multiply all the prime factors of the larger number by those prime factors of the smallernumber that are not already included. (Note: this sounds different than our textbook;but if you think about it, it is really the same thing. We end up with prime factors fromeach number and use each prime factor the number of times it appears the most).Example: Find the Lowest Common Multiple of 240 and 924.Solution: Write the prime factorization of each. 924 = 2 x 2 x 3 x 7 x 11 and240 = 2 x 2 x 2 x 2 x 3 x 5, therefore the lowest common multiple is:(2 x 2 x 3 x 7 x 11) x (2 x 2 x 5) = 924 x 20 = 18,480
- 24. improper fractions.Examples: Mixed Fractions81937167321 ororSimplifying Fractions`The key to working with fractionsis to express them as simply aspossible. It is easiest to work with thesmallest possible numerator (north –up) and denominator (down). Thismakes it easier to visualize andperform mathematical operations. Afraction is in its lowest terms whenboth the numerator and denominatorcannot be divided evenly by anynumber except 1.Equivalent Fractions. If thenumerator and denominator of anyfraction are multiplied by the samenumber, the fraction looks differentbut really represents the same part ofthe whole. We know that any singlenumber multiplied by 1 is equal to thenumber we started with.Note that any fraction that has thesame numerator and denominator isequal to one – this is what we reallyhave when we multiply the numeratorand denominator by the same number.We have equivalent fractionswhen we express a fraction in 2different ways, both of which can besimplified to the same lowest terms.We also create an equivalent fractionwhen we divide the numerator anddenominator by the same number – inthis case, we are expressing the samefractions with smaller numbers.Examples: Equivalent Fractions10584634221====Simplest TermsKeeping fractions easy to look at,visualize, and work with is important.That is what we are doing when wesimplify fractions. If we can divideboth the numerator and denominatorby the same number (just another wayof expressing 1), we reduce thenumbers used to express that fraction.This makes it simpler to work with.When we reduce the numeratorand denominator by dividing each bythe same number and cannot reduce itMr. Breitsprecher’s Edition March 3, 2005 Algebra Connections, Page 2any further, we have expressed thatfraction in simplest terms.Examples: Simplest Terms4333129=÷5412126048=÷Common DenominatorsIf we are comparing 2 differentfractions (which is larger and which issmaller), we need to think of eachfraction in terms of the same, equalparts of the whole (denominator).When we add or subtract fractions, weneed to work with fractions that areexpressed with the same denominator– a common denominator.Rewriting fractions with thelowest common denominator is just aspecial example of writing anequivalent fraction – in this case, wefind equivalent fractions that share thesame denominator.Finding the Common DenominatorHere is a step-by-step method forfinding common denominators thatwill work every time:1. Examine the fractions – can wesee by observation what thecommon denominator is? If so –skip steps 2 and 3, goimmediately to step 4. The morewe practice working withfractions, the easier this stepbecomes.2. If we did not see anythingobvious in step 1, determinewhich fraction has the largestdenominator.3. Check to see if the smallerdenominator divides into thelarger one evenly. If so, move onto step 4. If not, check multiplesof the larger denominator untilyou can find one that the smallerdenominator can divide intoevenly. If you remember ourearlier lesson about lowestcommon multiples – that processwill take you immediately to thelowest common denominator(LMC=LCD).4. Write the 2 fractions asequivalent fractions with thecommon denominator.Note that we can always find acommon denominator by multiplyingthe denominators together – while thisworks, it can result in working withlarge numbers. Once this number isfound, go to step 4. If you cancomfortably work with these largernumbers, this method can be efficient.Examples: Common Denominators146&1472273&772173&21××=3015&3016151521&2215821&158××=Improper Fractions & MixedFractionsWhile working with an improperfraction is no different than workingwith a mixed fraction, most of us willfind it easier to interpret our results ifwe convert the improper fraction to amixed fraction (whole number &proper fraction).Divide the numerator (north = up)by the denominator (down) – thenumber of times the denominator goesevenly into the numerator is our“whole” number. The remainder, orparts that are left-over, are thenumerator. Now we have are left witha fraction that represents less than oneand a whole number for everythingelse.921911=Note: 11 divided by 9 is 1with a remainder of 3.1024010402=Note: 402 divided by 10 is 40with a remainder of 2.744732=Note: 32 divided by 7 is 4with a remainder of 4.
- 25. Graph of (3, 4)Quadrant IQuadrant IIQuadrant III Quadrant IVMr. Breitsprecher’s Edition November 10, 2005 Web: www.clubtnt.org/my_algebraA rectangular coordinatesystem is a plane with vertical andhorizontal number lines thatintersect at their 0 coordinate. Thevertical line is referred to as they-axis; the horizontal line, the x-axis. We call the point ofintersection the origin (0, 0).We graph or plot an orderedpair by locating the corresponding xand y values on our rectangularcoordinate system. To plot x=3 andy=4 (3, 4); start at the origin andmove right 3 and then up 4.The rectangular coordinatesystem is divided into quadrants.The quadrants are:Quadrant x-axis y-axisI Positive PositiveII Negative PositiveIII Negative NegativeIV Positive NegativeAn ordered pair is a solution ofan equation in 2 variables ifreplacing the variables with thecoordinates of the ordered pairresults in a true statement.If we are given one coordinateof an ordered pair solution, the othervalue can be determined bysubstitution. For example: x-4y=16Start by assuming x equalssome convenient number towork with, say 0 (0, y). Bysubstitution, we have:0-4y = 16-4y=16(-4y)/-4 = 16/-4y = (-4)Our ordered pair is (0, -4)A linear equation in twovariables is an equation that can bewritten in the form Ax+By=C,where A and B are are not both 0.We call the form Ax+By=Cstandard form.To graph a linear equation intwo variables, find three orderedpairs that are solutions for theequation. Two points (each anordered pair) determine the line.We use the third point as our“check.” When we plot the threepoints, a straight line should connectall three points.Graphing Linear Equations: InterceptsAn intercept point of a graph is the point where the line crosses an axis. The x-intercept is the point where a line crosses the x-axis. If that point is some number, let’scall it “a,” then the x-intercept is “a” and the corresponding intercept point is (a, 0). Ifa graph intersects the y-axis at a point we call “b,” then b is the y-intercept and thecorresponding intercept point is (0, b). To find intercept points:x-intercept point is determined by letting y=0 and solving for xy-intercept point is determined by letting x=0 and solving for xExample: 5x + 2y = 10We find the x-intercept bysetting y = 0 and solvingfor x.5x + 2(0) = 105x=10(5x)/5 = 10/5x = 2The x-intercept is (2, 0)Next, we will find they=intercept by settingx = 0 and solving for y.5(0) + 2y = 102y = 10(2y)/2 = 10/2y = 5The y-intercept is (0, 5)
- 26. Mr. Breitsprecher’s Edition November 10, 2005 Algebra Connections, Page 2Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Linear Equations: SlopeThe slant or steepness of a line is refered to as slope. Theslope (m) of a line passing through points (x1, y1) and (x2, y2)can determined by:rise change in y y2 – y1m = run change in x x2 – x1Example: Assume a line passes through (2, 1) and(4, 4).Solving for m, we get:4 - 1m = 4 - 23m = 2Graphing this line, we can see how the slope indicates howmany units the line’s rise (y2 – y1) and run (x2 – x1)It makes no difference what 2 points of a line we choose tofind its slope. The slope of a line is the same everywhere onthe line. Note that when we calculate the slope, it makes nodifference what points we assign as (x1, y1) and (x2, y2) as longas we make sure that when we call one point x1, we use itscorresponding y-coordinate as point y1. A positive slope goesup; a negative slope goes down (from left to right).A horizontal line has a slope of 0, because the “rise” iszero and its run is infinite – by definition, performing thisdivision results in 0.A vertical line has a slope that is undefined, because the“rise” is infinite and its run is 0 – by definition, division by 0 isundefined.Nonvertical parallel lines have the same slope. Twononvertical lines are perpendicular if the slope of one is thenegative reciprocal of the slope of the other.Linear Equations: Intercepts(Continued from page 1)Special Case: Graph of x = c is a vertical line withx-intercept of “c.” In this case, c = 3.Special Case: Graph of y = c is a vertical line with x-intercept of “c.” In this case, c = 3.
- 27. Mr. Breitsprecher’s Edition November 14, 2005 Web: www.clubtnt.org/my_algebraA graph of a straight line is calleda linear equation in 2 variables.They can always be written in the formAx+By = C where A and B are notBOTH 0. Note that a linear equationmay appear to have only one variable(x or y). This means that the othervariable has a coefficient of 0. We callthe form Ax+By = C standard form.A linear equation defines arelationship between 2 variables, x andy, and is called a relation. The set ofall x-coordinates is called the domainof the relation. The set of all y-coordinates is called the range of therelation.A function is a set of orderedpairs that assigns each x-valueprecisely one y-variable. In otherwords, each x-value predicts a y-value.When setting up linearequations, it is important establishthe relationship so that each value ofy is dependent on the value of x.This results in each value of y beinga “function” of x.Function notation uses thesymbol f(x) means function of x. forexample: f(x) = 3x-7.f(-1) = 3(-1)-7 = -10In other words, f(-1) means tosubstitute that value (-1) for x anddetermine the corresponding y-value.Because function notation meansthat each x-value has precisely 1 y-value, there can only be 1 value of yfor each value of x. In other words, avertical line can always be drawnthrough and it will only intersect alinear equation 1 time – this is calledthe vertical line test. A functioncannot curve back on itself or containany type of angle that results in a valueof x to “predict” more than 1 y.Three Useful Forms for LinearEquation in 2 VariablesWe will use 3 forms to representlinear equations in 3 variables:1. Standard Form: Ax+By = C,where A and B are not both 0.2. Slope-Intercept Form: y = mx+b,where m is the slope of the line andb is the y-intercept (0, b).3. Point-Slope Form: y-y1 = m(x-x1),where m is the slope and x1, y1 is apoint on the line.Note that standard form is themost general equation – recall that themultiplication property of equalitytells us we can multiply BOTH SIDESof an equation in standard from bysome number and not change theidentity of the equation (line).In other words, A, B, and C willchange and each will still be valid.For this reason, standard form haslimited practical applications.The slope-intercept form,however, will always have uniquevalues of m (slope) and b (x-intercept)for each line.What Does Slope MeanSlope tells us the slant or tilt of a line – it is defined as rise over run and can beexpressed as:rise change in y y2 – y1m = run change in x x2 – x1Note that when we take an equation in standard form (Ax+By = C) and solve it fory, the resulting coefficient of x is the slope – THIS IS NOT THE SAME AS SAYINGTHAT” A” IS THE SLOPE – IT CERTAINLY IS NOT!Please see the examplebelow:10x-5y = (-5)-10x+10x-5y = (-5)–10x-5y = (-10)-10x(-5y)/(-5) = [(-5)-2x]/(-5)y = 1+2xy = 2x+1Note that we nowhave a slope of 2, whichis not the coefficient Afrom our standard form.When we solve any equation in standard form, the resulting coefficient of x isactually the slope of the line. Please see the examples above.
- 28. Important DefinitionsRectangular Coordinate System. Plane with vertical and horizontal number lines thatintersect at their 0 coordinate.X-axis. The number horizontal lineY-axis. The number vertical lineOrigin. Where the lines cross, represented by the point x=0, y=0Ordered Pair. A pair of coordinates, one representing “x” and one representing “y.” Byconvention, written in the format of (x,y).Relation. Set of ordered pairsDomain. Set of all x-coordinatesRange. Set of all y-coordinatesFunction. Set of ordered pairs that assigns to each x-value exactly one y-valueVertical Line Test. If a vertical line intersects a graph more than once, the graph is not afunctionOnline ResourcesThe Coordinate Plane and Graphing Linear Equations. In this unit well be learningabout equations in two variables. A coordinate plane is an important tool for workingwith these equations. http://www.math.com/school/subject2/lessons/S2U4L1GL.htmlTutorial: Graphing Linear Equations. When you graph linear equations, you will endup with a straight line. Lets see what you can do with these linear equations.http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut21_graph.htmGraph of a Line. We can graph the linear equation defined by y = x + 1 by findingseveral ordered pairs. For example, if x = 2 then y = 2 + 1 = 3, giving the ordered pair (2,3). Also, (0, 1), (4, 5), (-2, -1), (-5, -4), (-3, -2), among many others, are ordered pairs thatsatisfy the equation. http://www.algebra-online.com/graph-lines-1.htmGraphs of Linear Equations: Lines and Slope. Lessons, some practice setshttp://www.math.com/school/subject2/lessons/S2U4L1GL.htmlGraphing Inequalities in 2 Variables. The solution set for an inequality in twovariables contains ordered pairs whose graphs fill an area on the coordinate plane called ahalf-plane. An equation defines the boundary or edge of the half-plane.http://www.algebra-online.com/graphing-inequalities-1.htmWorksheets: Linear Equations and Inequalities. Need more practice? Theseworksheets will help. http://www.edhelper.com/LinearEquations.htmThinkQuest: Graphine Linear Equations and Inequities. Let’s review basics ofgraphic equations and inequalities. http://library.thinkquest.org/10030/6gleai.htmCreate Graphs ofEquations and InequalitiesAutomatic Graph Solutions: Equations. Enter the equation you want to plot, in termsof the variables x and y, set the limits and click the Plot button.http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=graphs&s2=equations&s3=basicAutomatic Graph Solutions: Inequalities. Enter the polynomial inequality you wantto plot, in terms of the variables x and y, set the limits and click the Plot button.http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=graphs&s2=inequalities&s3=basicLikewise, the point-slope formwill always have a unique value for m(slope) and x1, y1 must be points onthe line.Graphing Linear Inequalities inTwo VariablesA linear inequality is two variables isan inequality that can be written inone of the following forms:Ax+By < CAx+By ≤ CAx+By > CAx+By ≥CThe solution set for an inequalityin two variables contains orderedpairs whose graphs fill an area on thecoordinate plane called a half-plane.An equation defines the boundary oredge of the half-plane.Graphing Linear Inequalities1. Graph the boundary line bygraphing the related equation.Draw the line solid if theinequality symbol is ≤ or ≥.Draw the line dashed if theinequality is < or >.2. Choose a test point not on theline. Substitute its coordinatesinto the original inequality.3. If the resulting inequality is true,shade the half-plane that containsthe test point. If the inequality isnot true, shade the half-pane thatdoes not contain the test point.Example: y-2x ≤ 1The boundary would be y =2x+1. Choose easy to work withvalues for x and find corresponding y-values, i.e. (0, 3), (1, 3), and (3, 7).Remember, 2 points determine a line.The third point is a “check.” Wewould graph this as a solid line; ourboundary is included in our half-plane.To determine which half-plane,represents our solution, choose a testpoint; say (0, 0). Substituting thisback in our original inequality:y-2x ≤ 1, we see 0-2(0) ≤ 1 resultsin 0 ≤ 1. This is a true statement,so we shade that half-plane.Mr. Breitsprecher’s Edition November 14, 2005 Algebra Connections, Page 2
- 29. Mr. Breitsprecher’s Edition January 18, 2005 FREE!Some of us have a hard timelearning algebra because we get tocaught up worrying about "the rightway" or "best way." The truth is thatthere are many ways to look atapplying algebra concepts.Textbooks are written to presentalgebra from a certain viewpoint - onthat "leads" students through thelessons. Sometimes, trying to look atan idea from a different point of viewis helpful.A good example of this is howtextbooks present Least CommonMultiple, which happens to be thesame thing as the Lowest CommonDenominator. In Beginning Algebraclass, K Elayn Martin-Gays text startsher presentation with factoring andbuilding the Least Common Multiple.She presents fractions and lowestcommon denominators next and wantsto start students thinking aboutfactoring early, she starts with a greatdeal of information about primefactorization. Another class or bookmight have empathized primefactorization later - students can findthe Least Common Multiple/LowestCommon Denominator without adetailed review of factoring.Which is better? Its six versus ahalf-dozen - you can decide foryourself. What is important is that wepractice procedures until we arecomfortable and reasonably accurate.For many students in BeginningAlgebra, the class will become easier ifwe practice our homework one-on-onewith someone that knows the material -your instructor or the Math Center atUW-W are great places to start.Textbooks and classroom lecturesemphasize a method that "leads"people to the next topic. Workingoutside of class, one-on-one or in smallstudy groups, allows us to focus onsharing DIFFERENT ways to look atprocedures - perhaps ways that aremore compatible with a personsstrengths or learning styles. Here are2 ways to look at a basic algebraic skillthat is necessary for success in acollege level algebra class.Point Counterpoint: LeastCommon Multiple = LowestCommon DenominatorMethod 1. The Least (or Lowest)Common Multiple (LCM) is thesmallest number that two or morenumbers will divide into evenly. TheLCM also represents the simplest, orLowest Common Denominator, whenwe need to add or subtract fractions.To find the LCM/LCD:1. Find the Greatest Common Factor(GCF) of the numbers or thelargest number that divides evenlyinto both numbers.2. Multiply the numbers together3. Divide the product of the numbersby the Greatest Common Factor.Example: Find the LCM of 15 and 121. Determine the Greatest CommonFactor of 15 and 12 which is 32. Either multiply the numbers anddivide by the GCF (15*12=180,180/3=60)3. OR - Divide one of the numbersby the GCF and multiply theanswer times the other number(15/3=5, 5*12=60)Method 2: Because we can onlyadd or subtract fractions if theyrepresent the same pieces of the whole(same denominators), finding thelowest common denominator (LCD) isimportant. The lowest commondenominator is also the least commonmultiple. Here is a short review onhow to find the LCD (which is theLCM), so that we can create equivalentfractions that maintain their originalidentify.1. Factor each denominator into itsprime factors2. Build the LCDa. Each unique prime factormust appear at least onceb. Raise factors to the highestpower it appears in any one ofthe original denominatorsExample: Find the lowest commondenominator between fractions thathave the denominators of 15 and 12.1. Prime factors of 15 = 3*5. Primefactors of 12 = 2*2*32. Using EACH prime factor (step2a) and raising each to the highestpower it appears in the originaldenominator, we get: 22*3*5 =60.Which of these 2 approaches is"best"? You will have to decide foryourself - note that our textbookpresents the process in a mannersimilar to Method 2. Findingalternative ways to learn can behelpful. Sometimes, working withanother person is the best way to lookat your options - it is often the onlyway that someone can see the best wayto "teach" a complex process,procedure, or idea. Many believe thatthis "self-understanding" is the realvalue of higher-level math classes.Isnt this really what a collegeeducation is all about?YOUDECIDEBasement of McCutchan Hall, Rm. 1; Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.
- 30. In order to understand linearequations in 1 variable, recall that avariable is a number that is notidentified. It is often represented by"x" or "y," any letter can be used. Alinear expression is a mathematicalstatement that performs functions ofaddition, subtraction, multiplication,and division, but has no exponents(or powers) and no variables thatmultiply or divide each other. Someexamples of linear expressionsinclude:• x + 4• 2x = 4• 2x + yThe following examples ARENOT linear expressions:• x2• 2xy + 4• 2x / 4yA linear equation is amathematical statement that has anequal sign and linear expressions.Examples include:• 2x + 4 = 10 (linearequation in 1 variable)• 3x - 4 = -10 (linearequation in 1 variable)• 4x - 4y = 8 (linearequation in 2 variables)Solving linear equationsinvolves applying logical steps tosimplify the expressions while stillmaintaining the equality andoriginal identity or solution. Eachstep creates an equivalent equation,not a new one. Like mostprocedures, this will be easiest tolearn if we establish a sequence ofsteps and practice using them untilwe are comfortable.Most of us would agree thatmath problems are easier to workwith if we ELIMINATE ORCLEAR ALL FRACTIONS first.To SOLVE LINEAREQUATIONS IN ONEVARIABLE, practice the following6-step process:1. Multiply both sides of theequation to CLEARFRACTIONS if they occur –multiply both sides of theequation by the lowest commondenominator.2. Use the distributive property toREMOVE PARENTHESES ifthey occur. This will oftencreate “like terms.”3. Simplify both sides of theequation, COMBINE LIKETERMS.4. Get all terms containingvariables, all VARIABLETERMS ON ONE SIDE OFTHE EQUATION and allnumbers on the other side byusing the addition property ofequality.5. GET THE VARIABLEALONE by using themultiplication property ofequality to remove thecoefficients of any term with avariable.6. CHECK YOUR SOLUTIONby substituting it into theoriginal equation.Let’s look at how the examplesof linear equations in one variable,identified earlier, would be solvedby applying our set of logicalprocedures.We will not look at our 3rdexample of a linear equationbecause it is a linear equation in 2variables.Example 1: 2x + 4 = 101. Note that there are no fractions,so we will go to step 2.2. There are no parentheses toremove, so we will go to step3.3. There are no like terms tocombine, so we will go to thenext step, 4.4. Isolate "x" to one side of theequation with the additionproperty of equality. Add theopposite, a –4, on both sides ofthe equation to move that termto the other side: 2x + 4 - 4 =10 – 4. This simplifies to 2x =6. We now have only the termwith the variable on the left,but it has a coefficient.Mr. Breitsprecher’s Edition February 9, 2005 Web: www.clubtnt.org/my_algebraAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.
- 31. 5. Isolate the variable (remove thecoefficient) by using themultiplication property ofequality. Multiplying bothsides by 1/2. Now we have:2x*(1/2) = 6*(1/2). This willsimplify to x=3.6. Check your answer in theoriginal equation, 2x+4 = 10, bysubstituting our proposedsolution (x=3) for the variable.(2*3)+4 = 10. This simplifiesto: 6+4 = 10. This is a truestatement; we have a solution tothis linear equation in onevariable.Example 2: 3x - 4 = -101. Note that there are no fractions,so we will go to step 2.2. There are no parentheses toremove, so we will go to step 3.3. There are no like terms tocombine, so we will go to thenext step, 4,4. Isolate "x" to one side of theequation with the additionproperty of equality. Add theopposite, a 4, on both sides ofequation to move that term tothe other side: 3x - 4 + 4 = -10+ 4. This simplifies to 2x = -6.We now have only the termwith the variable on the left, butit has a coefficient.5. Isolate the variable (remove thecoefficient) by using themultiplication property ofequality. Multiplying bothsides both sides by 1/3. Nowwe have: 3x *1/3 = -6*1/3.This will simplify to x = -26. Check your work in theoriginal equation, 3x-4 =-10, bysubstituting our proposedsolution (x=-2) for the variable.(3*-2)-4 = -10; -6-4 = -10. Thisis a true statement and we havea solution to this linear equationin one variable.Give Me Life, Liberty, And The Pursuit OfAlgebraic Properties Of EqualityWhether solving simple algebraic expressions, linear equations, or a host of other types ofproblems, applying the basic properties of real numbers is the key to success. Using thesetools, we can work towards isolating variables and solving for the unknown. Here is aquick review of some of the algebraic properties that are used to simplify linear equations.Transitive Property of Equality. Property indicates the logic behind equalities and howit can create inferences about equalities. If a = b and b = c, then a = c.Commutative Property of Addition. Numbers can be added in any. a + b = b + aAssociative Property of Addition. When adding three numbers, it doesn’t matter if thefirst two or the last two numbers are added first. (a + b) + c = a + (b + c)Commutative Property of Multiplication. Numbers can be multiplied in any order. Forexample: a * b = b * aAssociative Property of Multiplication. When multiplying three numbers, it doesn’tmatter if the first two or the last two numbers are multiplied first. (a * b) * c = a * (b * c)Distributive Property. Property indicating a special way in which multiplication isapplied to addition of two (or more) numbers. For example: a(b + c) = a*b + a*cAddition Property of Equality. If you add the same number to each side of an equation,the two sides remain equal. If a = b, then a + c = b + c. If a = b, then a + c = b + c.Multiplication Property of Equality. If each side of an equation is multiplied by thesame number, then the two sides remain equal. 3 = 2 + 1, then 3 x 4 = (2+1) x 4.And if a = b, then ac = bc. If a = b, then a * c = b * c.Online ResourcesEarlier this year, Algebra Connections published some Internet resources for some ofthese skills. Here are more interactive tutorials and reviews. Sorry for the long URL’s.If you have any problems, just enter the domain name (http://domain_name.org)WITHOUT the directories (delete everything after the first forward slash /). Then followthe links on that page to the sites listed, using the directories as a guide. Remember, thereare never spaces in a URL, please use the underscore where indicated.Addition Property of Equalityhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut12_addeq.htmMultiplication Property of Equalityhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut13_multeq.htmSolving Linear Equations (Algebra Lab – great site!).http://www.algebralab.org/practice/practice.aspx?file=Algebra2_1-3.xmlSolving Linear Equationshttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut14_solve.htmSolving Linear Equationshttp://education.yahoo.com/college/student_life/math_homework/problem_list?id=minialg1gt_1_1Mr. Breitsprecher’s Edition February 9, 2005 Algebra Connections, Page 2Starducks Coffee is Harsh and Bitter -- Just Like Me.I need to keep my edge -- luckily there is a Starducks near by. One sip of their nasty, bitterbrew puts the grouch back in me. Its their blend of the harshest, oldest, and worst coffeebeans in the world. You will want to spit out your first taste -- but dont. Enjoy the feeling!
- 32. The following are examples ofinequalities:a < b a is less than ba ≤ b a is less than or equal to ba > b a is greater than ba ≥ b a is greater than or equal to bGraphing InequalitiesWhen we draw equalities on anumber line, we merely have to place adot or point on the location indicated.Graphing inequalities is not hard, but weneed to indicate the entire set of numbersthat are part of that inequality – it is notjust one point, there are infinite points.We will do this by using a “o” forthe point that is NOT included in ourgraph of an inequality. We will thendraw an arrow from that point to showwhich way the set of numbers thatrepresents that inequality extends.Example: x < 1|--|--|--|--o--|--|-3 -2 -1 0 1 2 3Example: x > -1|--|--o--|--|--|--|-3 -2 -1 0 1 2 3Remember, EVERY inequalityrepresents a SET of numbers, to the leftor right of where we start.When we graph inequalities thatinclude “equal to” such as “greater thanor equal to” or “less than or equal to,”we need to INCLUDE the starting pointin our graph. We do this by using “ ”and drawing our arrow to represent theset from that point.Example: x ≤ 1|--|--|--|-- --|--|-3 -2 -1 0 1 2 3Example: x ≥ -1|--|-- --|--|--|--|-3 -2 -1 0 1 2 3In order to understand linearinequalities, recall that a variable is anumber that is not identified. It is oftenrepresented by "x" or "y," any letter canbe used.A linear expression is amathematical statement that performsfunctions of addition, subtraction,multiplication, and division, but has noexponents (or powers) and no variablesthat multiply or divide each other.A linear inequality is amathematical statement that has one ofthe inequality signs above and linearexpressions.The examples of linear equalitiesdiscussed so far are fairly straight-forward. The graphs will becomeintuitively obvious with practice. Wheninequalities are not so obvious, we canuse a similar process to that which isused to solve linear equations. Theproperties of equalities are similar to theproperties of inequality.Addition Property for InequalitiesIf a < b, then a + c < b + cIn other words, adding orsubtracting the same expression toboth sides of an inequality does notchange the inequality.Example: x – 5 ≤ 1x – 5 ≤ 1x – 5 + 5 ≤ 1 + 5x ≤ 6|--|--|--|-- --|--|2 3 4 5 6 7 8Recall that the product of 2 realnumbers with the same sign isALWAYS positive. Also recall thatthe product of 2 numbers withdifferent signs is ALWAYS anegative number.Multiplication Properties forInequalitiesWhen multiplying by a positivevalue, If a < b AND c ispositive, then ac < bcWhen multiplying by a negativevalue, If a < b AND c isnegative, then ac > bc.In other words, multiplying thesame POSITIVE number to bothsides of an inequality DOES NOTCHANGE the inequality.Multiplying the same NEGATIVEnumber to both sides of aninequality REVERSES the sign ofthe inequality.Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Mr. Breitsprecher’s Edition February 14, 2005 Web: www.clubtnt.org/my_algebra
- 33. Multiplication Properties forInequalities (continued)Example: 5x < -10(1/5) * 5x < -10 * (1/5)x < (10/5)x < 5|--|--|--|--|--|--o--|-1 0 1 2 3 4 5 7Example: (x / 3) > 93 * (x / 3) > 9 * 3[(3x) / 3] > 27x > 27|--o--|--|--|--|--|--|26 27 28 29 30 31 32 33In the next examples, we willmultiply BOTH sides of the equationby a negative number. T ISALWAYS MEANS we need tochange the direction of the inequalitysign.Example: (-3x) ≥ 9 Note: Thecoefficient of the first term is (-3).Therefore, we will multiply BOTHSIDES by (-1/3). The inequalityneeds to be reversed to make thestatement true.(-1/3) * (-3x) ≤ 9 * (-1/3)x ≤ -9/3x ≤ -3|--|-- --|--|--|--|–5 –4 –3 -2 -1 0 1If you are unsure about this, pleasechoose any number from oursolution and check it in our originalequation. You will get a “true”statement.Example: (-x/3) > 7. Note: Thecoefficient of the first term is (-1/3).Therefore, we will multiply BOTHSIDES by (-3). The inequalityneeds to be reversed to make thestatement true.-3 * (-x / 3) < 7 * -3x > -27|---o---|---|---|---|-28 -27 -26 -25 -24 -23Solving Linear Inequalities is the Same asSolving Linear Equalities, EXCEPT…Yes, if you have mastered keeping track of positive and negative signs when performingmultiplication (we have defined division in terms of multiplication), solving linearequations is the same WITH ONE VERY IMPORTANT EXCEPTION.Let’s look at an example and see if we can make this clear. Look at this statement, 6 > 3.Certainly, we would all agree that 5 is greater than 3.Now we will multiply both sides by a convenient number, let’s use –1; however, anynegative number would do. Is (-1)(6) still greater than (3)(-1)? NO!!!!!-6 is not greater than –1!Notice that it is left than –1 on the number line:|-- --|--|--|--|-- --|--|-7 –6 –5 –4 –3 –2 –1 0 1 2Notice that multiplying by a negative number changes the signA positive number multiplied by a negative number results in a negative number.A negative number multiplied by a negative number is a positive number.Multiplying BOTH the left and right side of an equation by a negative number is never anissue (equations contain an “=” sign). IT IS ALWAYS AN ISSUE when we multiplyBOTH the left and right side of an INEQUALITY by a negative number. WE HAVETO “FLIP,” (or change the direction ) OF THE SIGN! This is also true if you divideby a negative.Online ResourcesNote: These URL’s can be long and frustrating to enter. You should be able to go to thedomain name (domain_name.com) and find links to that will take you to these pages.Algebra: Solving Linear Inequalitieshttp://en.wikibooks.org/wiki/Algebra:Solving_linear_inequalitiesS.O.S. Math: Inqualitieshttp://www.sosmath.com/algebra/inequalities/ineq01/ineq01.htmlMath.com: Inequalitieshttp://www.math.com/school/subject2/lessons/S2U3L4DP.htmlMathematics Help Central: Solving Linear Inequalitieshttp://www.mathematicshelpcentral.com/lecture_notes/intermediate_college_algebra/solving_linear_inequalities.htmThinkQuest. Solving Linear Equationshttp://library.thinkquest.org/C0110248/algebra/ielinearsolving.htmMr. Breitsprecher’s Edition February 14, 2005 Algebra Connections, Page 2Express
- 34. Multiplying and dividingrational expressions is just likemultiplying and dividing fractions –they just look more complicated.Recall that:0≠=× bdwherebdacdcba≠=×=÷ bdwherebcadcdbadcbaWhen working with fractions,we can perform the arithmetic andthen look at our results to see if it isin “lowest terms.” Many of us, withpractice, can intuitively see how tosimplify fractions because werecognize when numbers containcommon factors. We are able to“see” the common factors withoutfactoring the numerators anddenominators.Also notice that dividingfractions is simply multiplying bythe inverse. Once we arecomfortable multiplying fractions,we will quickly learn to divide themif we are able to remember torewrite the division asmultiplication. Instead of dividing,multiply by the inverse.Multiplying Rational ExpressionsMultiplying rationalexpressions works exactly likefractions, but the expressions aremore complicated. Not only are weworking with numerators anddenominators, each are apolynomial. We need a way to keepthe polynomials in the numeratorsand denominators manageable.Few of us will intuitively seeall common factors in rationalexpressions. The key will be tocompletely factor the numeratorsand denominators FIRST, thencancel out common factors (which isapplying the FundamentalPrincipal of RationalExpressions). The factors that areleft represent our answer and thisanswer is expressed in its simplestform.Do you see that we DO NOTACTUALLY PERFORM ANYMULTIPLICATION WHEN WEMULTIPLY RATIONALEXPRESSIONS?We factor, which means writerational expression as a product, andthen cancel out all common factorsthat appear in BOTH the numeratorand denominator (FundamentalPrincipal of Rational Expressions).What is left after factoring andcanceling out common factorsbetween the numerator anddenominator IS OUR PRODUCT!Steps in Multiplying RationalExpressionsStated formally, as in a typicalAlgebra Textbook, the process formultiplying rational expressionsthat we have been discussing can bewritten as:1. Factor numerators anddenominators.2. Multiply Numerators andmultiply denominators.3. Write the product inlowest terms (applyFundamental Principal ofRational Expression,PR/QR = P/Q, where R isa nonzero polynomial)Note that we can apply thesesteps BY COMBINING 1 & 2 byfactoring the numerators andnominators and rewriting each ofthese sets of factors as 1 rationalexpression that contains ALLfactors from each numerator anddenominator.This is how AlgebraConnections will present itsexamples. We will completelyfactor each rational expression,writing the factors as one rationalexpression, keeping factors of thenumerators on top and factors of thedenominators on the bottom.Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Mr. Breitsprecher’s Edition April 4, 2005 Web: www.clubtnt.org/my_algebra
- 35. Mr. Breitsprecher’s Edition April 4, 2005 Algebra Connections, Page 2Example 1:xyyyx31059×yxyyyx635)52()33(=××××=Example 2: 353421538yxzzxy×−35342)3()53()222(yxzzxy×××××−=Example 3:166212722−×+++xxxxx82)4(2)4)(4()]3(2[)]4()3[(−=−=+−×+×+××+=xxxxxxxxxDividing Rational ExpressionsJust like fractions – once we arecomfortable with the process ofmultiplying rational expressions, weare ready to divide if we rememberto rewrite our division asmultiplication.Instead of dividing, multiply bythe reciprocal. As we have justseen, that means to factorcompletely, cancel out commonterms that appear in BOTH thenumerator and denominator, andwrite down the remaining factors.Note that we can write divisionof rational expressions with the “÷”sign or with a fraction bar.Example 4:xxxx65356535÷=Either way, we invert thedivisor and multiply.Example 4:xx 6535÷21253)32(55635==×××=×=xxxxExample 5: )272(2xx÷422121252727xxxxx=××=×=Example 6:252651222−−−÷++xxxxx235)23)(12[()5()]5)(5[()12(262551222−−=−+×+−+×+=−−−×++=xxxxxxxxxxxxxIndicated Division with FractionBarMany beginning algebra books,such as K. Elayn Martin-Gay’sIntroductory Algebra, do notpresent division of rationalexpressions with fraction bars – theyintroduce this concept as “complexfractions.”Remember, fractions alwaysindicate division. Thedenominator is the divisor. Theresult of division is called thequotient. Regardless of how wewrite division (fraction bar or “÷”),we need to invert the divisor (itsreciprocal) and rewrite the problemas multiplication.Example 7:613ba +babababa222)(13)32()(163+=+=×××+=×+=Example 8:31212−−xx)1(23)]1)(1[(13212−××+−=−×−=xxxxxExample 9:2352+a652135 22+=×+=aaOnline ResourcesMultiplying Rational Expressionshttp://www.purplemath.com/modules/rtnlmult.htmhttp://www.algebra-online.com/multiplying-rational-expressions-1.htmDividing Rational Expressionshttp://www.purplemath.com/modules/rtnlmult2.htmhttp://www.algebra-online.com/dividing-rational-expressions-1.htmMultiplying & Dividing RationalExpressionshttp://www.sci.wsu.edu/~kentler/Fall97_101/nojs/Chapter7/section1.htmlhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut32_multrat.htmhttp://faculty.ed.umuc.edu/~swalsh/Math%20Articles/RationalE.htmlMultiplying, Dividing, Adding,and Subtracting RationalExpressionshttp://tutorial.math.lamar.edu/AllBrowsers/1314/RationalExpressions.aspFactoring a Polynomial:3. Are there any common factors? If so,factor them out.4. How many terms are in thepolynomial:a. Two terms: Is it the difference of2 squares? a2-b2= (a-b)(a+b)1. See if any factors can befactored further. Watch fordifference of 2 squares!Quick Review: Two ImportantExponent Rulesam*an= am+nam/an= am-na. Three terms: Try one of thefollowing patternsi. a2+2ab+b2= (a+b)2ii. a2-2ab+b2= (a-b)2iii. Otherwise, try to use anothermethod.b. Four terms: Try factoring bygrouping.
- 36. howMr. Breitsprecher’s Edition February 22, 2005 Web: www.clubtnt.org/my_algebraMultiplying polynomials is justanother application of thecommutative and associativeproperties of multiplication.Combined with the multiplicationrules for exponents, we have all thetools necessary to practice andmaster this algebraic skill.The key will be to apply theseprocedures in an organized,simplified fashion. Understandingthis process is important – timespent becoming familiar with theseprocedures of multiplyingpolynomials will repay itself.Like many things – it isprobably best to look at and practicesimpler examples before we move tothose that require a greaterunderstanding. Recall that apolynomial is an expression withone or more terms added to orsubtracted from each other. Forexample, x³ + 5x² - 8 is apolynomial.The simplest form of apolynomial only has 1 term – it iscalled a monomial. Let’s start bylooking at multiplying monomialsby monomials.Example 1: MultiplyingMonomials:(4x3)(3x4)• Group coefficients and likebases:(4*3)*( x2x4)• Add exponents and simplify:12x7Example 2: MultiplyingMonomials:(-3c4d)(2c2d3e)• Group coefficients and likebases:(-3*2)(c4c2)(dd3)(e)• Simplify:-6c6d4eMultiplying a Polynomial by aMonomialThis isn’t new; we looked atthis earlier without using theseterms. Recall that distributiveproperty: a(b+c) = ab + bc. Do yousee that this is really multiplying apolynomial by a monomial?Example 1: Multiplying aBinomial by a Monomial:4t(2t-3)• Multiply each term of thebinomial by 4t:4t(2t-3)• Apply the distributive property:(4t)(2t) + 2t(-3)• Simplify each term:8t2-6tSpecial Case Product FormulasThere are patterns in numbers and in algebra – it is important to see theserelationships. In some cases – multiplying 2 binomials takes on a special pattern. Wewill not discuss why they work or “prove” them; you can apply foil to see for yourself.Please practice these patterns so that you can recognize and apply them – they areextremely important when factoring polynomials.• Difference Of Squares: (a+b)(a-b) = a2-b2• Perfect Square Trinomials:(a+b)2= a2+2ab+b2(a-b)2= a2– 2ab + b2More Links for Polynomial Expressionshttp://www.ifigure.com/math/algebra/algebra.htmSolve Your Math Problem (you supply the problem, it provides the solution)http://www.webmath.com/polymult.htmlHow to Add, Subtract, Multiply, and Divide Polynomialshttp://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Polynomial.htmlInteractive Exponents and Polynomial Review (requires plug-in, available at site)http://www.mathnotes.com/Intro/aw_introchap3.htmlPractice Test: Polynomial Concepts and Operations (interactive)http://www.mccc.edu/~kelld/polynomials/polynomials.htm
- 37. Example 2: Multiplying a Trinomialby a Monomial:-4a2(-3a2+ 2a – 3)• Multiply each term of the trinomialby –4a2:–4a2(-3a2+ 2a – 3)• Apply the distributive property:(-4a2)(-3a2) + (-4a2)(2a) + (-4a2)(-4)• Simplify:12a4– 8a3+ 16a2So we have seen how to multiplypolynomials by monomials – this wasjust an application of the distributiveproperty. Apply the same principleswhen multiplying polynomials withmore than 1 term (i.e. binomial,trinomials, etc.).THE KEY IS TO REMEMBERTO MULTIPLY EACH TERM OFTHE FIRST POLYNOMIAL BYEACH TERM OF THE SECONDPOLYNOMIAL. We can draw arrowsto help us keep track – this is especiallyuseful when multiplying polynomialsthat contain many terms.Mr. Breitsprecher’s Edition February 22, 2005 Algebra Connections, Page 2Example: MultiplyingPolynomials With More Than 1Term:(x+5)(x+3)• Multiply each term in thesecond polynomial by eachterm in the first:(x+5)(x+3)• Apply the distributionproperty:(x*x)+(x*3)+(5*x)+(5*3)• Simplify:x2+3x+5x+15)• Combine like terms:x2+8x+15Note how the product of twobinomials (example above) equalsthe sum of the products of the firstterms, the outer terms, the innerterms and the last terms.The acronym FOIL (FirstOuter Inner Last) can be used tohelp memorize how to multiply 2binomials. In this author’s opinion– that is just an extra rule tomemorize; we don’t need it. Pleaseuse it if you find it helpful and beaware that you will probably see itin another math class.We can accomplish the samething be remembering to multiplyeach term of the first polynomial byeach term of the second polynomial.We can easily keep track of thisif we draw arrows for each productwhile we work out the solution.This approach works whenmultiplying any polynomial withmore than one term by any otherpolynomialUsing FOIL will ONLY workwhen multiplying two binomials. ItWILL NOT work whenmultiplying a binomial by atrinomial or with any other type ofpolynomial. If you have beensuccessful accurately using FOIL inthe past – please continue to use it.Sometimes, new approaches arehelpful – drawing arrows to keeptrack of each term as you multiplypolynomials, can be a convenientmethod. This author recommendskeeping memorization to aminimum when learning algebra.DefinitionsMonomial has one term: 5y or -8x2or 3.Binomial has two terms: -3x2+ 2, or 9y- 2y2Trinomial has 3 terms: -3x2+ 2 +3x, or9y - 2y2+ yDegree Of The Term is the exponent ofthe variable: 3x2has a degree of 2.Degree of a Polynomial is the highestdegree of any of its terms.When the variable does not have anexponent -understand that theres a 1 .One thing you will do whensolving polynomials is combine liketerms. Understanding this is the keyto accurately working withpolynomials. Let’s look at someexamples:• Like terms: 6x + 3x - 3x• NOT like terms: 6xy + 2x - 4The first two terms are like and theycan be combined:5x2+ 2x2– 3Combining like terms, we get:7x2– 3Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Report• Helium was up, feathers weredown.• Paper was stationery.• Fluorescent tubing was dimmedin light trading.• Knives were up sharply.• Pencils lost a few points.• Hiking equipment was trailing.• Elevators rose, while escalatorscontinued their slow decline.• Light switches were off.• Mining equipment hit rockbottom.• Diapers remain unchanged.• Shipping lines stayed at an evenkeel.• The market for raisins dried up.• Caterpillar stock inched up a bit.• Sun peaked at midday.• Balloon prices were inflated.• Kleenex nosed up.
- 38. Although you may not realizeit, you use the set of integers ineveryday mathematical calculations.Positive integers are all the wholenumbers greater than zero: 1, 2, 3, 4,5, ... .Negative integers are all theopposites of these whole numbers: -1, -2, -3, -4, -5, … . We do notconsider zero to be a positive ornegative number. For each positiveinteger, there is a negative integer,and these integers are calledopposites. For example, -3 is theopposite of 3, -21 is the opposite of21, and 8 is the opposite of -8. If aninteger is greater than zero, we saythat its sign is positive. If an integeris less than zero, we say that its signis negative.Integers are useful in comparinga direction associated with certainevents. Suppose I take five stepsforwards: this could be viewed as apositive 5. If instead, I take 8 stepsbackwards, we might consider this a-8. Temperature is another waynegative numbers are used. On acold day, the temperature might be10 degrees below zero Celsius, or -10°C. (note: we used Celsius, notFahrenheit, because Celsius has anabsolute zero, water freezes at 0;Fahrenheit uses 32 for the samepoint).The Number LineThe number line is a linelabeled with the integers inincreasing order from left to right,that extends in both directions(Please see top heading):For any two different places onthe number line, the integer on theright is greater than the integer onthe left.Example:9 > 4, 6 > -9, -2 > -8, and 0 > -5Absolute Value of an IntegerThe number of units a numberis from zero on the number line. Theabsolute value of a number isalways a positive number (or zero).We specify the absolute value of anumber n by writing n in betweentwo vertical bars: |n|.Examples:|6| = 6|-12| = 12|0| = 0|1234| = 1234|-1234| = 1234Adding IntegersWhen adding integers of thesame sign, we add their absolutevalues, and give the result the samesign.(Continued on page 2)What are some good ways to explainnegative numbers:? Here are some “real-world” examples:Accounting. Say I borrow $5 from youand buy lunch since I forgot my wallet. Iowe you $5 so on my balance sheet I have-5. Later I get my wallet and pay you the$5 back so I can subtract my debit (a netcredit, subtracting a negative is a positive).So we have:-5 -(-5) = -5 +5 = 0, which is what I nowowe.Driving. You are driving with cruisecontrol set at 65mph (in a 65 zone, ofcourse), which we will call your referencespeed. You see a sign stating that you areentering a 55 zone so you slow down 10mph ( -10). After a few miles a new signinforms you that you are entering a 65 zoneagain so you resume your original speed,thus removing (subtracting) the -10mphmodification. We thus have -10 - (-10) = 0,or no speed modification (thus you aremoving at the reference speed of 65 again).Books. You borrow 3 books from alibrary. You thus owe three books (-3).You read one and return (subtract) it (aborrowed book is a minus, thus a -1) andthus you have subtracted one book youowe, and now owe only two. And wehave:-3 -(-1) = -3 + 1 = -2Behavior. Johnny swears and fights a lot(two negatives). He feels he wants to getbetter so he decides to stop (thus removingor subtracting) fighting (a negative). Thushe now has -2 - (-1) = -2 + 1 = -1, or 1negative behavior he does a lot.Remember: Subtracting a negative isadding, a positive. Subtraction canALWAYS be though of as adding theopposite. If the greater of the 2 numbersis positive, so it the answer. If the greaterof the 2 numbers is negative, the answer isnegative. For example, (the opposite).Source: mathforum.orgPractice: Here’s an interactive Web:http://www.quia.com/fc/22424.htmlMr. Breitsprecher’s Edition January 26, 2005 Web: www.clubtnt.org/my_algebraWe get it now, Mr. “B.”Can we have some fish?
- 39. Adding Integers (from page 1)Examples:2 + 5 = 7(-7) + (-2) = -(7 + 2) = -9(-80) + (-34) = -(80 + 34) = -114When adding integers of theopposite signs, we take theirabsolute values, subtract the smallerfrom the larger, and give the resultthe sign of the integer with thelarger absolute value.Example:8 + (-3) = ?The absolute values of 8 and -3are 8 and 3. Subtracting the smallerfrom the larger gives 8 - 3 = 5, andsince the larger absolute value was8, we give the result the same signas 8, so 8 + (-3) = 5.Example:8 + (-17) = ?The absolute values of 8 and -17 are 8 and 17. Subtracting thesmaller from the larger gives 17 - 8= 9, and since the larger absolutevalue was 17, we give the result thesame sign as -17, so 8 + (-17) = -9.Example:-22 + 11 = ?The absolute values of -22 and11 are 22 and 11. Subtracting thesmaller from the larger gives 22 - 11= 11, and since the larger absolutevalue was 22, we give the result thesame sign as -22, so -22 + 11 = -11.Example:53 + (-53) = ?The absolute values of 53 and -53 are 53 and 53. Subtracting thesmaller from the larger gives 53 - 53=0. The sign in this case does notmatter, since 0 and -0 are the same.Note that 53 and -53 are oppositeintegers. All opposite integers havethis property that their sum is equalto zero. Two integers that add up tozero are also called additiveinverses.Subtracting IntegersSubtracting an integer is thesame as adding its opposite.Examples:In the following examples, weconvert the subtracted integer to itsopposite, and add the two integers.7 - 4 = 7 + (-4) = 312 - (-5) = 12 + (5) = 17-8 - 7 = -8 + (-7) = -15-22 - (-40) = -22 + (40) = 18Note that the result ofsubtracting two integers could bepositive or negative.Multiplying IntegersTo multiply a pair of integers ifboth numbers have the same sign,their product is the product of theirabsolute values (their product ispositive). If the numbers haveopposite signs, their product is theopposite of the product of theirabsolute values (their product isnegative). If one or both of theintegers is 0, the product is 0.Examples:In the product below, bothnumbers are positive, so we just taketheir product.4 × 3 = 12In the product below, bothnumbers are negative, so we take theproduct of their absolute values.(-4) × (-5) = |-4| × |-5| = 4 × 5 = 20In the product of (-7) × 6, thefirst number is negative and thesecond is positive, so we take theproduct of their absolute values,which is |-7| × |6| = 7 × 6 = 42, andgive this result a negative sign: -42,so (-7) × 6 = -42.In the product of 12 × (-2), thefirst number is positive and thesecond is negative, so we take theproduct of their absolute values,which is |12| × |-2| = 12 × 2 = 24,and give this result a negative sign: -24, so 12 × (-2) = -24.To multiply any number ofintegers:1. Count the number of negativenumbers in the product.2. Take the product of theirabsolute values.3. If the number of negativeintegers counted in step 1 iseven, the product is just theproduct from step 2, if thenumber of negative integers isodd, the product is the oppositeof the product in step 2 (givethe product in step 2 a negativesign). If any of the integers inthe product is 0, the product is0.Example:4 × (-2) × 3 × (-11) × (-5) = ?Counting the number ofnegative integers in the product, wesee that there are 3 negativeintegers: -2, -11, and -5. Next, wetake the product of the absolutevalues of each number:4 × |-2| × 3 × |-11| × |-5| = 1320.Since there were an odd numberof integers, the product is theopposite of 1320, which is -1320,so:4 × (-2) × 3 × (-11) × (-5) = -1320.Dividing IntegersTo divide a pair of integers ifboth integers have the same sign,divide the absolute value of the firstinteger by the absolute value of thesecond integer.To divide a pair of integers ifboth integers have different signs,divide the absolute value of the firstinteger by the absolute value of thesecond integer, and give this result anegative sign.Examples:In the division below, bothnumbers are positive, so we justdivide as usual.4 ÷ 2 = 2.In the division below, bothnumbers are negative, so we dividethe absolute value of the first by theabsolute value of the second.(-24) ÷ (-3) = |-24| ÷ |-3|=24÷3=8.In the division (-100) ÷ 25, bothnumber have different signs, so wedivide the absolute value of the firstnumber by the absolute value of thesecond, which is |-100| ÷ |25| = 100÷ 25 = 4, and give this result anegative sign: -4, so (-100) ÷ 25 = -4.In the division 98 ÷ (-7), bothnumber have different signs, so wedivide the absolute value of the firstnumber by the absolute value of thesecond, which is |98| ÷ |-7| = 98 ÷ 7= 14, and give this result a negativesign: -14, so 98 ÷ (-7) = -14.Source: www.mathleague.comMr. Breitsprecher’s Edition January 26, 2005 Algebra Connections, Page 2
- 40. Percents, like fractions anddecimals, are way to express how apart of something relates to a whole.If we talk about 1/2, .5, or 50% of apizza, we are talking about one of 2equal parts – half the pizza.Percentages describe therelationship to a whole divided into100 parts. The word “percent”means “out of 100.” The whole partis considered 100%. Do you seethat the percent sign (%) consists of2 zeros, as in 100?Percentages can also be used torepresent more than the wholeamount (100%). Assume we have100 pieces of candy. That wholerepresents 100%. Let’s assume webuy 50 more pieces of candy – 50parts of 100 equals 50%. Noticethat our original 100% + 50% more= 150% of what we started with.The advantage of working withpercentages is that we express eachpart of the whole in terms of 1/100.Percentages allow us to makemeaningful comparisons betweenparts of a whole, because they alwaysrefer to parts out of 100. If weconvert fractions to percentages (byperforming the indicated division andmultiplying by 100), we can noweasily see the relationship betweenfractions – even when thedenominators are different.A percentage is really just adecimal multiplied by 100. Most ofus will find it easier to work withpercentages, because we don’t have todeal with as many decimal places (i.e.12.5% = .125).The Three Percentage CasesTo explain the cases that arise inproblems involving percents, it isnecessary to define the terms that willbe used. Rate (r) is the number ofhundredths parts taken. This is thenumber followed by the percent sign.The base (b) is the whole on whichthe rate operates. Percentage (p) isthe part of the base determined by therate. In the example:5% of 40 = 25% is the rate (r)40 is the base (b)2 is the percentage. (p)There are three cases that usuallyarise in dealing with percentage, asfollows:Case I. Find the percentagewhen the base and rate are known, i.e.What number (p) is 6% (r) of 50 (b)?Case II. Find the rate when thebase and percentage are known, i.e. 20(p) is what percent (r) of 60 (b)?Case III. Find the base when thepercentage and rate are known, i.e. thenumber 5(p) is 25% (r) of whatnumber (b)?Source: http://www.tpub.com/math1/7a.htmHere are some common equivalents. Do you see the patterns when you relate onecolumn to another?Fraction Decimal PercentageTwentieths.........................................1/20......................05................................5%Tenths................................................1/10......................10..............................10%3/10......................30..............................30%7/10......................70..............................70%9/10......................90..............................90%Eighths.................................................1/8......................125......................... 12.5%3/8......................375......................... 37.5%5/8......................625......................... 62.5%7/8......................875......................... 87.5%Sixths ...................................................1/6......................16666..................16 2/3%5/6......................83333..................83 1/3%Fifths....................................................1/5......................2................................20%2/5......................4................................40%3/5......................6................................60%4/5......................8................................80%Quarters ..............................................1/4 .....................25..............................25%3/4 .................75.......................75%Thirds ..................................................1/3......................33333..................33 1/3%2/3......................66666..................66 2/3%Half ......................................................1/2 .....................5................................50%Mr. Breitsprecher’s Edition January 26, 2005 Web: www.clubtnt.org/my_algebra
- 41. Case I. 6% of 50 = ? The "of"has the same meaning as it does infractional examples, such as1/4 of 16 = 4In other words, "of" means tomultiply. Thus, to find thepercentage, multiply the base by therate. Of course the rate must bechanged from a percent to a decimalbefore multiplying can be done. Tofind the percentage, multiply theRate times base and then divideby 100.[6% (r) of 50 (b)]/100 = p(6 * 50)/100 = 3The number that is 6% of 50 is 3.To explain Case II and CaseIII, we notice in the foregoingexample that the base corresponds tothe multiplicand, the ratecorresponds to the multiplier, andthe percentage corresponds to theproduct.50 (b) * .06 (r) = 3.00 (p)Case II, ?% of 60 = 20.Recalling that the product dividedby one of its factors gives the otherfactor, we can solve the followingproblem:?% of 60 = 20We are given the base (60) andpercentage (20).60 (b) * ? (r) = 20 (p)We then divide the product(percentage) by the multiplicand(base) to get the other factor (rate).Percentage divided by base times100 equals rate. The rate is found asfollows:20/60 * 100 = 33 1/3 %Case III, 25% of ? = 5. Therule for Case II, as illustrated in theforegoing problem, is as follows: Tofind the rate when the percentageand base are known, divide thepercentage by the base and multiplyby 100.The unknown factor in Case IIIis the base, and the rate andpercentage are known.25% of ? = 5We are given the rate (25) andpercentage (5). To find the basewhen the percentage and rate areknown, divide the percentage by therate and multiply by 100.5/25 * 100 = 20Mr. Breitsprecher’s Edition January 26, 2005 Algebra Connections, Page 2Another Perspectives: Ratios, Proportions & PercentsCross-Products to the Rescue!!!Sometimes a different point of view is helpful – it gives us choices. Analternative way to think of percentages is to understand the relationshipbetween ratios, proportions, and percentages.Ratios tell how one number is related to another number. A ratio may bewritten as A:B or A/B or by the phrase "A to B". A ratio of 1:5 says that thesecond number is five times as large as the first. Percentages are really allratios – just remove the “%” sign and write the remaining number over 100.For example, 25% equals “25 to 100.” or the ratio 25/100.A proportion is a mathematical statement that two ratios are equal. If the tworatios are not equal, then it is not a proportion. An example would be theproportion that compares 3/6 = 4/8. Do you see that each is an equivalentfraction (in this example, 1/2). Please look at the proportion above andnotice that the product of the 2 outside terms equals the product of the 2inside terms, i.e. (3*8)=(4*6). This will always be true of any proportion.This is called a “cross-product.”If we remember that all percentages can be expressed as a ratio by removingthe “%” and placing the remaining number over 100, then we can use cross-products to solve the 3 types of percentage problems we have beendiscussion. Many will find this approach easier; because if we set up ourproportions correctly and understand how to algebraically solve for themissing variable, then we do not need to consider each of the three casesdiscussed in the preceding article. Cross-products will work for all three!Case I, 6% of 50 = ? Realize that we are solving for the percentage. 6%(rate or “r”) can ALWAYS be expressed as the ratio 6/100. The base is 50.We can use a variable (p) to represent the question mark (percentage) andwrite that ratio as p/50. ALWAYS express this ratio as the percentage overthe base. Now we have the following proportion: 6/100 = p/50. The crossproducts are (6*50) = (100*p), the product of the outside terms equals theproduct of the inside terms. This will simplify to 300= 100p. Solving for“p”, we get p=300/100 or 3. This is the same percentage that we calculatedearlier.Case II, ?% of 60 = 20. Realize that we are solving for the rate (r). It can beexpressed as the ratio as r/100 (always rate/100). We are given thepercentage (20) and the base (60). These are expressed as the ratio 20/60(always percentage over base). Now we have the following proportion:r/100=20/60. The cross products are (r* 60) = (100*20) When proportion isexpressed in this format, the cross-product is always the product of theoutside terms equals the product of the inside terms. This can be simplified,r(60) = 2000. Solving for rate (r), we get r = 2000/60 or 33 1/3%. This is thesame rate we calculated earlier.Case III, 25% of ? = 5. Realize that we are solving for the base. We canexpress the rate as the ratio 25/100 (always as rate/100). We are given thepercentage (and are solving for the base). This can be expressed as the ratio5/b (always percentage/base). Now we have the proportion of (25/100) =(5/b). The cross product is (25*b) = (100*5). When proportion is expressedin this format, the cross-product is always the product of the outside termsequals the product of the inside terms. Solving for b, we get b = (100*5)/25or 20. This is the same base we calculated earlier.
- 42. Mr. Breitsprecher’s Edition March 2, 2005 Web: www.clubtnt.org/my_algebraPolynomials are algebraicexpressions. They can contain realnumbers and variables. There cannotbe any division and square roots thatinvolve the variables. Only addition,subtraction and multiplication ofvariable terms are allowed.Polynomials contain more thanone term – that’s the meaning of theroot-word “poly” Think ofpolynomials as the sums ofmonomials.A monomial has one term:5y or -8x2or 3.A binomial has two terms:-3x2+ 2, or 9y - 2y2A trinomial has 3 terms:-3x2+ 2 +3x, or 9y - 2y2+ yThe degree of the term is theexponent of the variable: 3x2has adegree of 2.When the variable does nothave an exponent, it is assumed tobe 1. (i.e. 1x has a degree of 1).Example:x2- 7x - 6Each part, or “chunk” ofmathematical information is a termand x2is referred to as the leadingterm.CoefficientsA constant used to multiply anotherquantity or series is called acoefficient. Examples include 3x andax. In these 2 cases, 3 and a arecoefficients of x.Examples:Term Coefficientx21-7x -7-6 -6Examples of Expressions andPolynomials• 8x2+ 3x -2 is a polynomial withthree terms. It is a trinomial.• 8x-3+ 7y -2 is NOT a Polynomial,because the exponent is negative.• 9x2+ 8x -2/3 is NOT aPolynomial; it cannot havedivision.• 7xy is a Monomial. There is only1 term.Polynomials are usually written indecreasing order of terms. The largestterm or the term with the highestexponent in the polynomial is usuallywritten first. The first term in apolynomial is called a leading term.When a term contains an exponent, ittells you the degree of the term.DegreesDegree of a Term. Each term in apolynomial has a “degree,” or theexponent of that term. A term writtenPolynomial Definitions of Terms:• A monomial has one term: 5y or -8x2 or 3.• A binomial has two terms: -3x2 + 2, or 9y - 2y2• A trinomial has 3 terms: -3x2 + 2 +3x, or 9y - 2y2 + y• The degree of the term is the exponent of the variable: 3x2 has a degree of 2.• When the variable does not have an exponent -understand that theres a 1 e.g., 3xWorking With Polynomials: Just Collect Like Terms!One thing you will do when solving polynomials is combine like terms.Understanding this is the key to adding and subtracting polynomials. Let’s look atsome examples:• Like terms: 6x + 3x - 3x• NOT like terms: 6xy + 2x - 4The first two terms are like and they can be combined: 5x2+ 2x2– 3, combining liketerms, we get: 7x2– 3.Adding and Subtracting PolynomialsJust like working with polynomials, they key is to identify like terms. To addpolynomials, you must clear the parenthesis, combine and add the like terms. In somecases you will need to remember the order of operations. Remember, when adding andsubtracting like parts, the variable never changes.
- 43. Adding Polynomials (continued)Here are a couple of examples of how to add polynomials:(5x + 7y) + (2x - 1y)= 5x + 7y + 2x - 1y (Clear the parenthesis)=5x + 2x + 7y - 1y (Combine the like terms)= 7x + 6y (Add like terms)Another Example:(y2 - 3y + 6) + (y - 3y 2 + y3)y2 - 3y + 6+ y - 3y2 + y3 (Clear the parenthesis)y3 + y2 - 3y2 - 3y + y + 6 (Combine the like terms)y3 - 2y2 - 2y + 6 (Add like terms)Subtracting PolynomialsTo subtract polynomials, you must change the sign of terms being subtracted, clear theparenthesis, and combine the like terms. Heres an example:(4x2 - 4) - (x2 + 4x - 4)(4x2 - 4) + (-x2 - 4x + 4) (Change the signs)4x2 - 4 + -x2 - 4x + 4 (Clear the parenthesis)4x2 -x2 - 4x- 4 + 4 (Combine the like terms)3x2 - 4xAnother Example:(5x2 + 2x +1) - ( 3x2 – 4x –2 )5x2 + 2x +1 - 3x2 + 4x +2 (Change the signs and clear theparenthesis)5x2 - 3x2 + 2x+ 4x+1 + 2 (Combine the like terms)2x2+ 6x +3Online Tools & ResourcesAddition and Subtraction of Polynomialshttp://www.mathnotes.com/Intro/Hchapter3/aw_InterActt3_4.html (requires plug-in)Online Self-Check Quiz: Polynomial Basicshttp://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-02-825326-4&chapter=9&lesson=4Online Self-Check Quiz: Adding and Subtracting Polynomialshttp://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-02-825326-4&chapter=9&lesson=5Downloadable And On-Line Exercises And Review Materialhttp://www.mathmax.com/prealg/chapter/bk7c10.htmlAdding and Subtracting Polynomials Tutorialhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut27_addpoly.htmPolynomial Basics (.pdf format – quiz with solutions)http://www.clc.mnscu.edu/kschulte/online%20worksheet/Polynomial%20Basic.pdfAdding and Subtracting Polynomials (.pdf format – quiz with solutions)http://www.clc.mnscu.edu/kschulte/online%20worksheet/Polynomial%20Basic.pdfwithout an exponent has a degree of1 (assumed exponent of 1)Degree of a Polynomial. Eachpolynomial also has a degree. Thedegree of a polynomial is thegreatest degree of any term in apolynomial (largest exponent of any1 term).Example: Three TermPolynomial (Trinomial)6x2- 4xy + 2xy This three termpolynomial has a leading term to thesecond degree. It is called a seconddegree polynomial and oftenreferred to as a trinomial.Example: Four Term Polynomial9x5- 2x + 3x4– 2. This 4 termpolynomial has a leading term to thefifth degree and a term to the fourthdegree. It is called a fifth degreepolynomial.Evaluating PolynomialsThe value of a polynomialdepends on the replacement valuethat is used for each variable. Whenwe evaluate a polynomial, wemerely substitute the replacementvalue we are given for thevariable(s) and determine a solutionbased on those values.Notice that if we expresspolynomials in by writing themfrom their largest term to theirsmallest, and if we identify thedegree of each term AND the degreeof the polynomial (greatest degreeof any term), then we can easily seeif there are any like terms thatshould be combined.Online ResourcesPolynomial Basicshttp://www.purplemath.com/modules/polydefs.htmLike Termshttp://www.math.com/school/subject2/lessons/S2U2L4DP.htmlAdding/Subtracting Polynomialshttp://www.purplemath.com/modules/polyadd.htmExponents and Polynomialshttp://www.mathnotes.com/Intro/aw_introchap3.html (interactive,requires plug-in - available at site)More Polynomial Linkshttp://mathforum.org/library/topics/polynomials/Mr. Breitsprecher’s Edition March 2, 2005 Algebra Connections, Page 2
- 44. Mr. Breitsprecher’s Edition January 24, 2005 FREE!If you learn these properties, theywill help you solve problems in algebra.Lets look at each property in detail, andapply it to an algebraic expression.Commutative Properties. Thecommutative property of addition saysthat we can add numbers in any order.The commutative property ofmultiplication is very similar. It saysthat we can multiply numbers in anyorder we want without changing theresult.Addition5a + 4 = 4 + 5aMultiplication3 x 8 x 5b = 5b x 3 x 8Associative Properties. Bothaddition and multiplication can actuallybe done with two numbers at a time. Soif there are more numbers in theexpression, how do we decide whichtwo to "associate" first? The associativeproperty of addition tells us that we cangroup numbers in a sum in any way wewant and still get the same answer. Theassociative property of multiplicationtells us that we can group numbers in aproduct in any way we want and still getthe same answer.Addition(4x + 2x) + 7x = 4x + (2x + 7x)Multiplication2x2(3y) = 3y(2x2)Distributive Property. Thedistributive property comes into playwhen an expression involves bothaddition and multiplication. A longername for it is, "the distributive propertyof multiplication over addition." It tellsus that if a term is multiplied by terms inparenthesis, we need to "distribute" themultiplication over all the terms inside.2x(5 + y) = 10x + 2xyEven though order of operationssays that you must add the terms insidethe parenthesis first, the distributiveproperty allows you to simplify theexpression by multiplying every terminside the parenthesis by the multiplier.This simplifies the expression.Identity Property. The identityproperty for addition tells us that zeroadded to any number is the numberitself. Zero is called the "additiveidentity." The identity property formultiplication tells us that the number 1multiplied times any number gives thenumber itself. The number 1 is calledthe "multiplicative identity."Addition5y + 0 = 5yMultiplication2c × 1 = 2cInverses. Two numbers that arethe same distance from 0 on the numberline but lie on opposites of zero arecalled “opposites” or “additiveinverses.” Also, for each real number,a, there exists another number, 1/a. Theproduct of these number equals 1.(Note: the fraction a/a simplifies as 1.)Often, the term “reciprocal is used.Addition7 + -7 = 0Multiplication3 * 1/3 = 1Source: www.math.comIn algebra, we often work withexpressions. Simplify expressionsin the following order. If groupsymbols are present, simplifyexpressions within those first,starting with the innermost set. Also,simplify the numerator and thedenominator of a fraction separately.Then:1. Simplify exponentialexpressions.2. Multiply or divide in order fromleft to right3. Add or subtract in order fromleft to right.Try it - You’ll Like It!Here are some examples – Ihave kept the numbers “small,” sowe can just focus on the order ofoperations.1. 9÷3+4*7-20÷5=2. 23-[(12-3)-5]=3. 25-{15-[3(4-2)-6]}=4. (16+4)-(18-13)=5. (8+3)(14-6)=6. 8+11*3-4=7. 2{6[(5-2)-2) +4]=8. 27-14÷2=9. 5+3[4+(8-5)]=10. 52(4-3)*2+16=11. 3(4+2)-{7-[4-(6+5)]}=12. 10-(8+2)+3(6-4)=13. 2{6[(5-2)-2]+4}=14. 25-{15-[3(4-2)-6]}=15. 5+[7-(2+1)]=16. [11-(9-5)]-4=17. 2{5+3)[14-(6+2)]}=18. 18-{5+[14-(6+2)]=19. -2+9-(-9)*(-2)-6*(-2)=20. 3[4+(3-1)]+5[12-(8+2)]=
- 45. Symbols & Sets ofNumbersBeginning Algebra will be much easier if westart with an agreement of definitions andsymbols as they relate to numbers. It won’t beany fun to go back and review this as we moveforward, so here are some concepts that will“pop-up” from time to time in our class.Set. A collection of objects, called elements ormembers, enclosed in braces {a,b,c}Natural Numbers. {1,2,3,4...}Whole Numbers. {0,1,2,3,4...}Integers. {...,-3,-2,-1,0,1,2,3...}Rational Numbers.{real numbers that can beexpressed as a quotient of integers}Irrational Numbers. {real numbers that cannotbe expressed as a quotient of integers}Real Numbers. {all numbers that correspond toa point on the number line}Number Line. A line used to picture numbers(please see test examples on page 23)Absolute Value. The distance between a and 0on the number line. Expressed as |a|All semester, we will work with the followingsymbols. They are presented here for the samereason; it will be much easier to get anagreement of them now.= is equal to≠ is not equal to> is greater than< is less than≤ is less than or equal to≥ is greater than or equal toOrder Property of Real Numbers. For any 2real numbers a and b, a is less than b it a is tothe left of b on the number line.Examples-3<00>-30<2.52.5>0I remember a teacher explaining these symbolsto me as “the mouth of a whale.” Because thewhale is hungry, it always is about to eat thelarger of the 2 numbers; the whale has alreadyeaten the smaller number and is still hungry!-3 < 0Does MS Excel Know the Order of Operations?YES!!!Spreadsheets are powerful mathematical tools. Not only will it organizeand keep track of data, but it can perform a variety of operations. Usersmay either “insert” a variety of preset functions or create their ownformulas. To let Excel know you expect it to “do math” (create aformula) you need start your cell with an equal sign (=).Addition, plus sign (+) = 5+2 result 7Subtraction, hyphen (-) = 5-2 result 3(also used for negative) = -5 result -5Multiplication, asterisk (*) = 5*2 result 10Division, slash (/) = 5/2 result 2.5Power, chevron (^) = 5^2 result 25There are several operands to use for logic comparisons:Greater than, greater than sign (>) =5>2 result TRUELess than, less than sign (<) =5<2 result FALSEEqual to, equal sign (=) =5=2 result FALSENot equal to, Greater & Less than signs (<>) =5<>2 result TRUEMicrosoft Excel respects the Order of Operations and performs them inthe following sequence:ParenthesisPowersMultiplication and DivisionAddition and SubtractionThis means with an equation such as =5+3*2, Excel will do themultiplication 3*2 before it does the addition. The result would be 11. Ifyou wanted the addition to happen first, you would have to useparentheses such that =(5+3)*2, giving you a result of 16.Unlike traditional math, you should not use the square brackets, such as[(5+3)*(4-2)] for separations, ONLY use parenthesis, such as=((5+3)*(4-2)), result 16.Try it yourself! Using the information provided, can you create aformula that will “check” our results? Here are some examples of“formulas” using the Excel’s syntax. If you follow these examples fromthe exercise we looked at earlier, you can use Excel to verify that youBOTH understand the Order of Operations!Order of Operations: Examples of Excel Formulas1. =9/3+4*7-20/52. =23-((12-3)-5)3. =25-(15-(3*(4-2)-6))4. =(16+4)-(18-13)5. =(8+3)*(14-6)6. =8+11*3-47. =2*(6*((5-2)-2)+4)8. =2*(6*((5-2)-2)+4)9. =5+3*(4+(8-5))10. =5^2*(4-3)*2+1^611. =3*(4+2)-(7-(4-(6+5)))12. =10-(8+2)+3*(6-4)13. =2*(6*((5-2)-2)+4)14. =25-(15-(3*(4-2)-6))15. =5+(7-(2+1))16. =(11-(9-5))-417. =2*(5+3*(14-(6+2)))18. =18-(5+(4*(5-2)-7))19. =-2+9--9*-2-6*-2--420. =3*(4+(3-1))+5*(12-(8+2))
- 46. howMr. Breitsprecher’s Edition March 30, 2005 Web: www.clubtnt.org/my_algebraA rational number is a numberthat can be written as a quotient ofintegers. A rational expression isan expression that can be written asa quotient, or the form: P/Q, whereP and Q are polynomials and Q doesnot equal 0.Examples of rationalexpressions include:1564732359353,812222−−++−−−++−xxorxxxoraaaorxxThe last example above couldbe expressed as 4x2-6x-5, apolynomial.Like all real numbers, dividingby the integer 1 maintains the sameidentity or value. Dividing by 1turns our polynomial into a rationalexpression.Rational expressions are alsoreferred to as rational functions,because the identity or value of theequation is a result (or function) ofthe number(s) we substitute for thevariable(s).The set of values of realnumbers that can be substituted intoa function and result in a realnumber is called the domain.Values that cause a rationalexpression to be undefined (thedenominator becomes 0) are not partof the domain. Recall that we neverdivide by zero.Undefined Rational ExpressionsWith rational expressions(rational functions), we need towatch out for values that cause ourdenominator to be 0, an undefinedvalue. When looking for the domainof a rational function (the set ofvalues that result in a real number),find the values that cannot be used,values that make the denominator 0.Example 1: For what values of x isthe rational expression undefined?723+−yyTherefore we are solving:2y+7 = 02y+7-7 = 0-72y = -7(2y)/2 = -7/2y = -7/2For the value –7/2, this rationalexpression is undefined – in otherwords, the domain of this rationalexpression is every value of yEXCEPT –7/2. The denominator ofa fraction can never be equal to 0.Any number divided by zero isundefined – it is not a real number.Any value(s) of x which causes thedenominator to equal 0 cannot bepart of the domain of a rationalexpression.Example 2. For what values of x arethe rational expression undefined?23222+−+xxxWe are solving: x2-3x+2=0The first thing we must do isfactor the denominator:x2-3x+2 = (x-2)(x-1)Now we set each of thosefactors equal to 0 and solve:x-2 = 0 x-1 = 0x-2+2 = 0+2 x-1+1 = 0+1x = 2 x = 1We see that when x = 2 or 1, wehave a denominator of 0. Thisrational expression is undefinedwhen x = 2 and x = 1.Simplifying a Rational ExpressionRecall that we defined afraction as being in “simplest” formor “lowest terms” when thenumerator and denominator have nocommon factors other than 1 or –1.We also need to simplify rationalexpressions.A rational expression, P/Q(Q≠0) is in its simplest form orlowest term if the greatest commonfactor of its numerator anddenominator is 1.Rational expressions represent areal number for each value of thevariable that does not make thedenominator zero. All properties ofreal numbers apply to rationalexpressions.We use the fundamentalproperty of rational expressions towrite a rational expression in lowestterms.Make sure all of youralgebra expressions“make sense!”Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.
- 47. Online Resources: Rational ExpressionsInteractive Algebra Reviews: Rational Expressionshttp://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/tut_alg_review/framesA_4.htmlhttp://www.mathnotes.com/Combined/aw_comboch8.htmlUniversity Of Utah Online Rational Expression Reviewhttp://www.math.utah.edu/online/1010/rational/Short online quizzes for rational expressionshttp://library.thinkquest.org/11771/english/hi/math/tests/alg/5.htmlhttp://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-825083-8&chapter=11&lesson=1&headerFile=4&state=txRational Expressions: Simplifyinghttp://www.purplemath.com/modules/rtnldefs.htmhttp://www.jamesbrennan.org/algebra/rational/simplifying_rational_expressions.htmRational Expressionshttp://www.marlboro.edu/academics/study/mathematics/courses/f04emls/rational.htmlhttp://www.edteach.com/algebra/rational/simplifying_rational_expressions.htmhttp://www.tcc.fl.edu/dept/acsu/mc/docs/pdf/algebra/smrtexpr.pdfhttp://www.csun.edu/~hfmth006/Math094/094b/mod9/mod9ch1.pdfhttp://www.bestquest.com/printmaterial/AS_15_2_BlackLineMasters.pdfFundamental Principle of RationalExpressionsThe Fundamental Principle ofRational Expressions is that if P/Q is arational expression and R is a nonzeropolynomial, then:QpQRPR=Simplifying a rational expression issimilar to simplifying a ratio or fraction.Factor the numerator and thedenominator. Look for common factors,and eliminate them using theFundamental Principal of RationalExpressions (common factors cancelout).Just like with fractions, when werewrite a rational expression without thecommon factors in the numerator anddenominator, it is in simplest form.Example: Write each expression inlowest term1253225)32(322)32(5332225327230=××=××××=××××××=Mr. Breitsprecher’s Edition March 30, 2005 Algebra Connections, Page 2Note that this is an applicationof the fundamental property.Likewise:aaaaaaaaaaaaa7)2()2(727221432=××××=××××××=Note how we used the QuotientRule of Exponents whennumerators and denominators havethe same base.nmnmaaa −=Example: Simplify 54918aaaaaaaaa 222918918 1545454===×= −−To Simplify A RationalExpression:1. Completely factor thenumerator and denominator2. Apply the fundamentalprinciple of rational expressionsto divide out common factors.Note: This is what the GreatestCommon Factor does.Example: Simplify 231010xxx−−2210)1()1(10xxxx=−−=Example: Simplify1681622+−−xxx44)4)(4()4)(4(−+=−−−+=xxxxxxPoints to RememberWhen simplifying rationalexpressions, the fundamentalprinciple applies to commonfactors, NOT common terms.xxxxxx 2)2( +=×+Note that, on the right, wecannot factor out another x from thenumerator and denominator. Theyare common terms, NOT factors.Also remember that we canalways factor out -1. That will benecessary when we have factors thatonly differ by signs. Factoring out a-1 lets us rewrite an expressionusing opposite signs.Example: Simplify25102522+−−xxx55)5)(5()5)(5(1)5)(5()5)(5(−+−=−−+−−=−−+−=xxxxxxxxxxExample: Simplify27612722−−++xxxx94)9)(3()3)(4(−+=−+++=xxxxxxRational Expression = FractionsRational expressions arequotients of two algebraicexpressions. They can bemanipulated like fractions. Thefollowing properties of fractionsapply to rational expressions:• For any real numbers a and b,where b ≠ 0: -a/b = a/-b = -(a/b)• For any real numbers a and b,where b ≠ 0: -a/-b = a/b• For any real numbers a, b and k,where b ≠ 0 and k ≠ 0:(a·k)/(b·k) = a/b• For any real numbers a and b,where a ≠ b: (a-b)/(b-a) = -1
- 48. I now clearlysee that it’s1.32 x 10-24gramsMr. Breitsprecher’s Edition October 6, 2005 Web: www.clubtnt.org/my_algebraScientific Notation is a way towrite very large or very small numbersin a readable manner. The system isbased on the powers of Base Ten.Once a person masters this procedure,it is easy to use.Scientists developed this methodmany years ago to provide them with amethod to save time and eliminatemistakes by representing numbers withfewer zeros or decimal placeholders.Numbers are converted toscientific notation by increasing thepower of ten by one for each place thedecimal point is moved. Someexamples follow.Scientific Notation Examples:Fraction = 1/1000Decimal = 0.001Scientific Notation = 1 x 10-3(Note: the raised -3 is called theexponent.)Fraction = 1/100Decimal = 0.01Scientific Notation = 1 x 10-2Number = 1/10Decimal = 0.1Scientific Notation = 1 x 10-1Number = 10Scientific Notation = 1 x 101Number = 100Scientific Notation = 1 x 102Number = 1000Scientific Notation = 1 x 103Remember, the zeros in theseexamples are “placeholders.” Think ofthe 0 in terms of money. Would yousooner have $1 or $100? You can seehow important adding 0’s are.Decimals and Scientific Notation:To represent very large or verysmall numbers, Scientific Notationuses one digit to the left of the decimalpoint. The number is written with 1digit to the left of the decimal point(>0,<10) and only those digits to theright that give meaningful informationare included (significant digits). Forexample, 1,300,200 would be writtenas 1.3002 x 106. Note how the last 2zeros are not significant – they aremerely placeholders.The power of ten (exponent) willshow you how many places thedecimal point moves. Positiveexponents mean a larger number; thedecimal is moved to the right.Negative exponents mean a smallernumber; the decimal is moved to theleft.The number 8.2x10-6written indecimal format would be 0.0000082because the decimal point was moved6 places to the left to form the decimal0.0000082.5.63x10-5represented in decimalnotation is 0.0000563 Note: Thedecimal point was moved 5 places.4.11 x 10-6= 0.000004112000 = 2x 1000 = 2 x 103631,000,000 = 6.31 x 108Give It A Try: Scientific NotationTo work with numbers containing many zeros (either very large or very small) in anefficient manner that reduces the chance of making an error, use Scientific Notation or“Powers of Ten.” Each value is written as a number between one and ten multiplied byten raised to some power (positive for large numbers, negative for small ones). Canyou write these numbers in scientific notation?17. 4,875,00018. 510,000,00019. 78420. 2,000,000,00021. 0.6922. 0.200523. 0.0024924. 0.0635625. 0.000009126. 0.0000007927. 0.98328. 0.0000100629. 2.2130. 55531. 1,47232. 29,00033. 30,800,00034. 1,153,000,00035. 31,000,000,000,000,00036. 3,000,000,000,000,000,00037. 0.0004238. 0.00018339. 0.1065940. 0.000002541. 0.00000012842. 0.0843. 0.0000000000744. 0.0006250445. 0.00062546. 0.04040447. 0.00051. 1,0002. 100,0003. 0.0014. 105. 0.0000016. 1,000,0007. 0.18. 19. 0.00000000110. 10,00011. 60,00012. 36,000,00013. 1,214,000,00014. 155,000,000,00015. 3,06016. 520,000
- 49. MS Excel Knows Scientific NotationLet’s Check Your Work With MS ExcelMost spreadsheet software will accommodate Scientific Notation – after all, lookingat all the zeros in large and small numbers, even in a computer format, is asking forerrors. In MS Excel, when numbers are entered into cells that have a General numberformat, Excel can only accommodate up to 11 digits, regardless of column width. If thelength of the number exceeds 11 digits, Excel automatically converts the number toscientific, or exponential, notation.Excel formats Scientific Notation differently, but the concept is the same –everything is based on a power of 10. Numbers that exceed 11 digits results in a numberlike 123456789098 being converted to 1.23457E+11. Note that in traditional ScientificNotation, this would be expressed as 1.23456789098 x 1011.Do you see the relationshipbetween MS Excel’s “Scientific Notation” and standard “Scientific Notation”?Note that MS Excel also rounds the number off in order to display it. It actuallydoes carry the full number for calculations, however. You cant change how Excelexpresses numbers, but, when necessary, you can usually convert the scientific notationto a text format that correctly displays the original data. For our purposes, however, theMS Excel number display is fine.To check your answers for the exercise given earlier, just open MS Excel and enterthe numbers that were given in column A. Be sure to use a GENERAL NUMBERformat. Select Column “A” and pull down the FORMAT menu. Choose CELLS andbe sure the NUMBER tab is selected. Use GENERAL, which should be the first choiceon the list that appears. Click OK.Once you are sure that the cells will display the GENERAL NUMBER format,enter the values from our exercise in the range A1:A47. Note how when you enter thenumbers that exceed 11 characters, Excel returns the number in its version of ScientificNotation, not the standard format that was reviewed in textbooks.If you see an entry that consists of “#########”, this means the number is too wideto fit in the width of that column. Using your mouse, point to the right line at the columnheading until you see a DOUBLE-ENDED arrow. Click on the mouse and “drag” thecolumn wide enough to display that number.In order to see the relationship between the original number in conventional decimalform and its value in scientific notation, let’s create a formula that copies the contentsfrom Column A to Column B. In cell B1, enter =A1 and then highlight the rangeB1:B47. Pull down the EDIT menu and select FILL. Another set of choices willappear, choose DOWN. This simple formula will just copy the formats from column Ainto column B.Next, we want to format column B so that it displays all values in its version ofScientific Notation. Highlight column B, pull down the FORMAT menu, and selectCELLS. Again, be sure the NUMBER tab is selected. Look at your choices and selectSCIENTIFIC – the dialog box will ask you to choose how many decimals to display.Excel will carry the full value for calculations, but only displays those decimalplaces you request. For our purposes, it will be fine to round everything that is displayedby accepting the “default,” which is 2 decimal places. Click OK.If you have correctly followed these directions, you will have all of the numbersfrom the previous exercise in column A, cells 1-47. In column B, you should haveExcel’s version of Scientific Notation. When the numbers are small, you should seenegative exponents (after the E) and when numbers are large, you will see positiveexponents (after the E). Note that Excel is always referring to a base of 10. In ourexample, you should see the following in rows 1 and 2, columns A and B:1000 1.00E+03 Means 1.00 x 103100,000 1.00E+05 Means 1.00 x 105Want Some More Help?Check Out These GreatWeb ResourcesAnd TutorialsChemTutorhttp://www.chemtutor.com/numbr.htm#sciFordham Prephttp://www.fordhamprep.com/gcurran/sho/sho/lessons/lesson25.htmNYUhttp://www.nyu.edu/pages/mathmol/textbook/scinot.htmlU of Michiganhttp://www.astro.lsa.umich.edu/users/garyb/Course/WWW/Scinot/scinot.htmlU of Marylandhttp://janus.astro.umd.edu/astro/scinote/Indiana Uhttp://carini.physics.indiana.edu/P105S98/Scientific-notation.htmlU Nebraska – Lincolnhttp://www-class.unl.edu/chem/HTML/lab_sylabus/lab_week_3/Scientific_Notation.htmlInstitute for Energy andEnvironmental Researchhttp://www.ieer.org/clssroom/scinote.htmlLoganhttp://members.aol.com/profchm/sci_not.htmlWidener College(Interactive tutorial to“check” your workhttp://science.widener.edu/svb/tutorial/scinot.html
- 50. Mr. Breitsprecher’s Edition September 21, 2005 Web: www.clubtnt.org/my_algebraA term is a constant or avariable in an expression. In theequation 12+3x+2x2=5x-1; 12, 3x,and 2x2the terms. On the left are12, 3x and 3x2; while the terms onthe right are 5x, and -1.Combining like terms is aprocess used to simplify anexpression or an equation by usingaddition/subtraction of thecoefficients of the terms. Considerthe expression below5 + 7By adding 5 and 7, you caneasily find that the expression isequivalent to 12What Does Combining LikeTerms Do?Algebraic expressions can besimplified like the previous exampleby combining like terms. Considerthe algebraic expression below:12x + 7 + 5xAs you will soon learn, 12x and5x are like terms. Therefore, thecoefficients 12 and 5 can be added.This is a simple example ofcombining like terms.17x + 7What are Like Terms?The key to using andunderstanding how to combine liketerms is to recognize what like termsare and seeing when you have a pairof terms that are alike.The following are like termsbecause each term consists of avariable, x, and a numericcoefficient.2x, 45x, x, 0x, -26x, -xEach of the following are liketerms because they are all constants.15, -2, 27, 9043, 0.6Each of the following are liketerms because they are all y2with acoefficient.3y2, y2, -y2, 26y2For comparison, below are afew examples of unlike terms.The following two terms bothhave a single variable with anexponent of 1, but the terms are notalike since different variables areused.17x, 17zEach y variable in the termsbelow has a different exponent,therefore these are unlike terms.15y, 19y2, 31y5Although both terms belowhave an x variable, only one termhas the y variable, thus these are notlike terms either.19x, 14xySource: http://www.algebrahelp.comCombining Like TermsIn an ExpressionConsider the expression below:5x2+ 7x + 2 - 2x2+ 7 + x2We will demonstrate how to simplify this expression by combining like terms. First,we identify sets of like terms. Both 2 and 7 are like terms because they are both constants.The terms 5x2, -2x2, and x2are like terms because they each consist of a constant times xsquared.The coefficients of each set of like terms are added. The coefficients of the first setare the constants themselves, 2 and 7. When added the result is 9. The coefficients of thesecond set of like terms are 5, -2, and 1. Therefore, when added, the result is 4.With the like terms combined, the expression becomes:9 + 7x + 4x2The process of combining like terms is used to simplify expressions (above example)and to make equations easier to solve. The equation, which we will be simplifying andsolving, is below:x + 3x + 7 = 42 + x – 12When combining like terms it is important to preserve the equality of the equation byonly combining like terms on one side at a time. We will simplify the left hand side first.The first step is to find pairs of like terms; the second step is to add. The x and 3x are liketerms, so they are added resulting in 4x. (HINT: when a variable such as x has nocoefficient, its coefficient is 1, so x is the same as 1x.) The 7 does not have a like term, soit is not changed. The equation now reads:4x + 7 = 42 + x – 12(continued on page 2)
- 51. Key Concepts: EquationsTerm. A number or product of anumber and variables raised topowers.Numerical Coefficient. Numericfactor in a term.Like Terms. Exact same variablesto exact same powers (order doesnot matterUnlike Terms. NOT same variablesto exact same powers (order doesnot matter)Combining Like Terms. First, addcoefficients of like terms andmultiply by common variablefactors. Then, simplify with thedistributive property if possible andcombine like terms.Linear Equation. Written inAx+B=C where A, B, and C are realnumbers and A does not equal zero.Equivalent Equations. Have samesolution.Addition Property of Equality.We can maintain equivalentequations by adding the same termto each side. i.e. if a=b, then (a+b) isequivalent to a+c=(b+c).Simplifying Equations. Combinelike terms on 1 or both sides ofequation BEFORE doing anythingelse. Then, put ALL terms with avariable on 1 side of the equation.Writing Equations. If the sum of 2numbers equals a 3rd number, thenwe can write that as a+b=c andsimplify when solving for a byrewriting as a=c-b.Multiplication Property of Equity.If a, b, and c are real numbers and cdoes not equal zero, then a=b andac=bc are equivalent equations.(Note: we define division in termsof multiplication, therefore, themultiplication property of equalityapplies for division as well).Writing Equations. The sum of 3consecutive numbers would bewritten a+(a+1)+(a+2). The sum of3 consecutive odd numbers wouldbe a+(a+2)+(a+4).The next step is to simplify the right hand side of the equation. This time there is noterm which can be added with x, but there are two constants, which are like terms. The 42and the -12 are added, resulting in 30. The equation now reads:4x + 7 = x + 30Combining Like Terms:A Second Equation ExampleThe next example equation is shown below. Solving this equation will require bothsimplifying multiple signs and combining like terms.-9 + 12x - 10 - 4x = 8x - 6x + 46 – (- 1)The first step to simplifying this equation is to simplify the double negative sign infront of the 1. The second negative sign cancels out the first one, so there are no signs left,meaning that the 1 is positive. Review the rules of multiple numbers with positive andnegative signs if this concept is unfamiliar to you. When this step is completed, theequation becomes:-9 + 12x - 10 - 4x = 8x - 6x + 46 + 1We will start combining like terms on the left side with -9, a constant. The only otherconstant on the left side is -10, so we can add the two together as shown below. The sumof -9 and -10 is -19, thus the equation becomes:-19 + 12x - 4x = 8x - 6x + 46 + 1Next we will add together 12x and -4x because they are like terms (x to the firstpower is the only variable in each). The resulting equation is shown below:-19 + 8x = 8x - 6x + 46 + 1Now that all like terms on the left side have been combined, we start working on theright side by adding the constants 46 and 1 to get 47.-19 + 8x = 8x - 6x + 46 + 1Then we add the 8x and -6x to get 2x. The resulting equation is:8x - 19 = 2x + 47Now, the equation can be solved using addition, subtraction, and division, followingthe rules for solving equations. Algebra Connections will review those steps in anotherissue.Combining Like Terms: Online HelpEquation Practice ProblemsEquations that require you to combine like terms before solving the equation.http://www.algebrahelp.com/lessons/simplifying/combiningliketerms/pgw1.htmEquation CalculatorWill automatically combine like terms and solve the equation while showing all requiredwork. (The equation calculator will not work with exponents.)http://www.algebrahelp.com/calculators/equation/calc.jspCombining Like Terms CalculatorSimplifies multiple signs and combines like terms in a given expressionhttp://www.algebrahelp.com/calculators/expression/calc.jsp .Mr. Breitsprecher’s Edition September 21, 2005 Algebra Connections, Page 2Academic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.
- 52. Mr. Breitsprecher’s Edition February 10, 2005 Web: www.clubtnt.org/my_algebraGeorge Polya was a Hungarianwho immigrated to the United Statesin 1940. His major contribution ishis work in problem solving. Hewas frustrated with the practice ofhaving to memorize information.He was a talentedmathematician and an excellentproblem solver. He created a 4-stepproblem-solving model that is usedin a variety of academic andpractical situations all over theworld.Step 1: Understand the problem.Sometimes the problem lies inunderstanding the problem. If youare unclear as to what needs to besolved, then you are probably goingto get the wrong results. In order toshow an understanding of theproblem, you, of course, need toread the problem carefully.Sounds simple enough, butsome people jump the gun and try tostart solving the problem before theyhave read the whole problem. Oncethe problem is read, you need to listall the components and data that areinvolved. This is where you will beassigning your variable.Step 2: Devise a plan (translate).When you devise a plan(translate), you come up with a wayto solve the problem. Setting up anequation, drawing a diagram, andmaking a chart are all ways that youcan go about solving your problem.Step 3: Carry out the plan (solve).The next step, carry out the plan(solve), is big. This is where yousolve the equation you came up within your devise a plan step. Thefollowing two properties of realnumbers are important tools thathelp simplify and solve equationsand inequalities.Addition Property ofEquality. If a = b, then a + c =b + cMultiplication Property ofEquality. If a = b, then a(c) =b(c)Step 4: Look back (check andinterpret).You may be familiar with theexpression dont look back. Inproblem solving it is good to lookback (check and interpret).Basically, check to see if you usedall your information and that theanswer makes sense.Always ask yourself, “Did Iactually answer the question?” Ifyour answer does check out, makesure that you write your final answerwith the correct labeling or units.Source: www.wtamu.eduStrategies for Solving Linear EquationsYou may ask, “Why are we back to solving linear equations?” The answer is because thatis what this unit on problem solving is really all about! Think of story problems asequations (mathematical statements that algebraic expressions are equal) or inequalities(mathematical statements that algebraic expressions are NOT equal).Our challenge is to translate the “story” from a word problem into a mathematicalstatement. Then we can apply algebraic procedures to determine a solution. In class wepresented a 6-step process to solve linear equations. Here, it is presented as a 4-stepprocess. Notice how Step 1, simplify each side, is actually steps 1-3 as presented in class.Some people may find a 4-step procedure more useful, others, a 6-step process.Step 1: Simplify each side, if needed.• Remove fractions (multiply BOTH SIDES by the lowest common denominator).• Remove parenthesis or groupings (apply the distributive property).• Combine like terms.Step 2: Get all terms with variables (variable terms) to one side of the equation (additionproperty -- add the opposite to BOTH SIDES). Move all numbers on the other side(addition properties -- add the opposite to BOTH SIDES).Step 3: Remove any values (coefficients) that are in front of the variable (multiplicationproperty – multiply BOTH SIDES by the reciprocal of the coefficient(s).Step 4: Check your answer by substituting the “proposed” solution back into the originalproblem. THIS IS AN IMPORTANT STEP, especially in a “Beginning Algebra” class.Please take advantage of the opportunity to review algebraic procedures, practice mathskills, and verify that the solution is correct. If you think about it, doesn’t it just makesense? Please maximize your practice and learn these procedures and skills byperforming ALL CHECKS!
- 53. Additionincreased bymore thancombined togethertotal ofsumadded toSubtractiondecreased byminus, lessdifference between/ofless than, fewer than72Multiplicationoftimes, multiplied byproduct ofincreased/decreased by a factor of (thisone is both addition or subtraction ANDmultiplication)Divisionper, aout ofratio of, quotient ofpercent (divide by 100)Equalsis, are, was, were, will begives, yieldssold forVocabulary"Per" Means "Divided By” as "I drove 90 mileson three gallons of gas, so I got 30 miles pergallon" Also 30 miles/gallon"A" Sometimes Means "Divided By” as in"When I tanked up, I paid $3.90 for three gallons,so the gas was 1.30 a gallon, or $1.30/gallon"Less Than” If you need to translate "1.5 lessthan x", the temptation is to write "1.5 - x".DONT! Put a "real world" situation in, and youllsee how this is wrong: "He makes $1.50 an hourless than me." You do NOT figure his wage bysubtracting your wage from $1.50. Instead, yousubtract $1.50 from your wage"Quotient/Ratio Of" Constructions If aproblems says "the ratio of x and y", it means "xdivided by y" or x/y or x÷y"Difference Between/Of" Constructions If theproblem says "the difference of x and y", it means"x - y"Want to see more? Check out purplemath’sTranslating Word Problems at this URL:http://www.purplemath.com/modules/translat.htmTranslating Word Expression intoAlgebraic Expressions“What is the sum of 8 and y?” ( 8 + y)“4 less than y” (y – 4)“y multiplied by 13” (13y)“The quotient of y and 3” (y / 3)“The difference of 5 and y” (5 – y)“The ratio of 9 more than y to y” [(y + 9) / y]“Nine less than the total of a number (y) and two” [(y + 2) – 9]or (y – 7)“The length of a football field is 30 yards more than its width.Express the length of the field in terms of its width” (y * y + 30)or (y2+30)“Twenty gallons of crude oil were poured into two containers ofdifferent size. Express the amount of crude oil poured into thesmaller container in terms of the amount y poured into the largercontainer." The expression theyre looking for is found by thisreasoning: There are twenty gallons total, and weve already pouredy gallons of it. That means that there are X gallons left.” (20 – y)“Twice as much as the unknown” (2 * x)“Two less than the unknown” (2 – x)“Five more than the unknown” (5 + x)“Three more than twice the unknown” (3 + 2x) or (2x+3)“A number decreased by 7” (x – 7)“Ten decreased by the unknown” (10 – x)“Sum of a number and 20” (x + 20)“Product of a number and 3” (x * 3)“Quotient of a number and 8” (x / 8)“Four times as much” (4 * x)“Three is four more than a number” (3 = 4 + x)“Sheris age (x) 4 years from now” (x + 4)“Dans age (x) 10 years ago” (x – 10)“Number of cents in x quarters” [x (.25)]“Number of cents in 2x dimes” [2x(.10)]“Number of cents in x+5 nickels” [(x + 5)(.05)]“Separate 17 into two parts” (x and 17 – x)“$20,000 separated into two investments” (x and 20,000 – x)“Distance traveled in x hours at 50 mph” (x * 0)“Distance traveled in 3 hours at x mph” (3 * x)“Distance traveled in 40 minutes at x mph” (2x / 3)Note: (40 Minutes = 2/3 hours)“Interest on x dollars for 1 year at 5%” (0.05x)“Two consecutive integers” (x and x + 1)“Two consecutive even integers” (x and x + 2)“Two consecutive odd integers” (x and x + 2)NOTE: No unit labels such as feet, degrees, and dollars are used in equations.In this book we have left these labels off the answers as well. Just refer to the"Let x =" statement to find the unit label for the answer.Mr. Breitsprecher’s Edition February 10, 2005 Algebra Connections, Page 2
- 54. Understanding what fractionsare is the key to being able to workwith them. The rules of adding,subtracting, multiplying, anddividing them are the same whetherwe are working with fractionsconsisting of numbers or thosecontaining variables.Adding/Subtracting FractionsAdding and subtractingfractions is not hard if we rememberto work with commondenominators. This should makesense. How can we makemeaningful combinations ofdifferent parts of the whole if we arenot talking about the same sizeparts?For example, we don’t speak ofadding 1 pint to a quart of milk. Weadd 1 pint to 2 pints and end up with3 pints. It would sound odd to talkabout a “quart and a pint” of milk.Likewise, we don’t add quarters andeighths without finding the commondenominator first.Be sure that BEFORE addingor subtracting fractions, we find aCOMMON DENOMINATOR.Here is a 3-step process foradding/subtracting fractions:1. Convert the fractions toequivalent fractions that havethe same number in thedenominator.2. Add or subtract the numerators.Do not do anything with thedenominator! Simply writeyour answer on the commondenominator.3. Simplify or reduce the fractionif possible.Example:7321+Note that the lowest commondenominator is 14. Actually, anycommon denominator will do – butif we work with the lowest commondenominator, we make simplifyingour answer easier. We can find acommon denominator bymultiplying the originaldenominators together. For moreon determining the LCD, pleaserefer to the article to the left.This means we will need tomultiply BOTH the numerator andthe denominator of the first fractionby 7. The numerator anddenominator of the second fractionwill need to be multiplied by 2.141314614722737721=+=×+×Mr. Breitsprecher’s Edition January 29, 2005 Web: www.clubtnt.org/my_algebraAcademic Support ServicesFREE Tutoring And Academic Support Services!!!Basement of McCutchan Hall, Rm. 1Mon-Thurs: 9 a.m. – 9 p.m.Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m.Lowest Common Denominators & Equivalent FractionsBecause we can only add or subtract fractions if they represent the same pieces of thewhole (same denominators), finding the lowest common denominator (LCD) isimportant. The lowest common denominator is also the least common multiple – thiswas reviewed earlier. Here is a short review on how to find the LCD (which is theLCM), so that we can create equivalent fractions that maintain their original identify.1. Factor each denominator into its prime factors2. Build the LCDa. Each unique prime factor must appear at least onceb. Raise factors to the highest power it appears in any one of theoriginal denominators3. Convert fractions INDIVIDUALLY to equivalent fractions with the new LCDby multiplying the numerator AND denominator by the missing factorsWhile this procedure may, at first, sound different than the process that was introducedearlier in the semester when we reviewed least common multiples, do you see it isreally just a simpler set of steps to accomplish the same thing. The least commonmultiple IS the lowest common denominator!
- 55. Example: −21158The lowest common denominatorhere is 30 – it happens to be theproduct of the 2 differentdenominators we started out with.This will always work, but when westart out with larger numbers in thedenominators, it becomes morecumbersome to work with.We will need to multiply BOTHthe numerator and the denominator ofthe first fraction by 2. The numeratorand denominator of the secondfraction will need to be multiplied by15.3013015301615152122158=−=×−×Note that when adding orsubtracting mixed fractions, we canadd/subtract each part by itself (wholenumber and fraction) and write theanswer together. The key is to be surethe fractional part has a commondenominator.Example:612321 +Adding the whole number part ofthese mixed fractions isstraightforward (1+2=3). To add thefraction part, we need to find acommon denominator – the lowestcommon denominator (LCD) is 12.We will need to multiply BOTHthe numerator and the denominator ofthe first fraction by 4. The numeratorand denominator of the secondfraction will need to be multiplied by2. Adding the fraction part (using thecommon denominator):65121012212822614432==+=×+×Now we can combine our 2 parts(whole and fraction) to get:653Mr. Breitsprecher’s Edition January 29, 2005 Algebra Connections, Page 2Multiplying FractionsIn many ways, multiplyingfractions is the easiest operation of thefour. There is no need to have acommon denominator. Be carefulwith mixed fractions, however.Before multiplying them, re-writethem as an improper fraction. Youcan always express your final answerback to a mixed fraction, if need be.Here is a 3-step process formultiplying fractions:1. Change all mixed fractions toimproper fractions2. Multiply the numerators togetherand multiply the denominatorstogether. Each results in thenumerator/denominator of youranswer.3. Simplify or reduce the fraction ifpossible.Example:2835667283or=×Dividing FractionsIf one understands how tomultiply fractions, dividing them isnot hard. Recall that we can definedivision in terms of multiplication –instead of dividing by a number; wecan multiply by the reciprocal. This istrue in EVERY case – but it isessential when dividing fractions.Here is a 3-step process for dividingfractions:1. Change all mixed numbers toimproper fractions.2. Flip the second fraction,placing the bottom number on top andthe top number on the bottom.3. Continue as with themultiplication of fractions.Example:92125÷815244529125or=×=When dividing fractions, note that wehave really just taken advantage of ourdefinition for division. Thinking ofdivision in terms of multiplication bythe reciprocal and thinking ofsubtraction in terms of addition of theopposite (change sign) can be useful.Note that addition andmultiplication share a special feature:order does not matter (commutativeand associative properties ofaddition/multiplication). This meansthat when we are adding ormultiplying fractions, we can workwith more than 2 at a time.Decimals = FractionsWe should close our review offractions by pointing out that decimalsare really just fractions where heindicated division has been performed.They are really the same as fractionswith denominators of 10, 100, 1,000,and so forth.The decimal point really indicatesthat we are dealing with a “part-of-a-whole” or a fraction. Because thedenominators of these fractions arealways a multiple of ten, it is easier toindicate this with a decimal point.Example:100311231.12 =More Fraction ResourcesFractions, Decimals, Percentages:Explanationshttp://descartes.cnice.mecd.es/ingles/3rd_year_secondary_educ/Fract_dec_ptges/Fracciones_1.htmSimplifying Fractionshttp://www.helpwithfractions.com/simplifying-fractions.htmlFraction Linkshttp://www.mathleague.com/help/fractions/fractions.htmReducing Fractions & LCDhttp://www.sparknotes.com/math/prealgebra/fractions/section2.rhtmlFractions Fast Factshttp://www.mccc.edu/~kelld/fraff.htmPowerPoint Presentation: Findingthe Lowest Common Denominatorhttp://www.ceres.k12.ca.us/iweb/lessons/Monicas%20Math/Fractions%20XII.pptPre-Owned Vehicles in First-Crash Condition!Why go anywhere else and gettaken advantage of?COME HERE FIRST!

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