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# Algebra Notes in Newsletter Form

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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!

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### Algebra Notes in Newsletter Form

1. 1. Algebra Newsletter NotesA creative way to presentconcepts, notes & study guidesin a newsletter format.Excellent Resource!www.clubtnt.org/my_algebra
2. 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
4. 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
6. 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. 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
9. 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. 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. 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?
13. 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