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    Unit 2 notes Unit 2 notes Document Transcript

    • Polynomials Notebook.notebook October 02, 2012 Unit 2­ Polynomials POLYNOMIALS At a birthday party, Crusty the Clown took the  age of the birthday girl and multiplied it by 6,  subtracted 2, multiplied it by 7, and then added  3.  The resulting number was 493.  How old is  the birthday girl? May 11­12:48 PM May 11­12:48 PM What is a polynomial??? A polynomial with 3 terms is called a trinomial. Monomial = polynomial with 1 term Binomial = polynomial with 2 terms A polynomial is made up of terms that are There is also a quadrinomial (4 terms) & a quintinomial (5 terms) but we rarely only added, subtracted or multiplied. use these words. May 11­12:58 PM May 14­7:59 AM The degree of a polynomial with only one variable is the Find the degree of each polynomial. largest exponent of that variable. 1. 3x2 The degree of this trinomial is 4. 2. 4x y ‐1 2 3 3. 3x2 +4x2y3 ‐ 5z4 If the polynomial has more than one variable, you need to look at each term. You find the degree of the polynomial by adding the exponents of each term and naming the largest power. The degree of this trinomial is 3. May 14­7:53 AM May 14­8:13 AM 1
    • Polynomials Notebook.notebook October 02, 2012 The standard form of a polynomial is to put the terms with the highest degree first. Quiz Time!!! Arrange the following in standard form: Just kidding...but here are a 1. 4x2 ‐ 2x + 3 ‐ 5x3 few questions... 2. ‐1 + x3 ‐ 9x2 + 8x May 14­7:55 AM May 14­9:58 AM May 14­8:32 AM May 14­8:34 AM Adding & Subtracting Polynomials May 14­10:43 AM May 14­10:48 AM 2
    • Polynomials Notebook.notebook October 02, 2012 May 14­10:48 AM May 14­11:12 AM May 14­10:56 AM May 14­11:11 AM May 14­10:49 AM May 14­11:14 AM 3
    • Polynomials Notebook.notebook October 02, 2012 May 14­11:14 AM May 14­11:15 AM Remember, to add polynomials simply combine like terms. To subtract polynomials, distribute the -1 and then combine like terms. May 14­10:40 AM May 14­10:39 AM May 14­10:39 AM May 14­11:23 AM 4
    • Polynomials Notebook.notebook October 02, 2012 10. 22. Homework: 30. 46. p. 113 #2- 58 eoe May 14­11:30 AM May 14­11:31 AM Answers to p. 113 #2­ 58 eoe WARM UP 2.  4 6.  3 10.  1 14.  ­10t + 8 Maya takes a number, multiplies it by 18.  ­5a b  ­ 5ab 22.  ­5x ­ 6 4, subtracts 5, multiplies by 4 again 26.  ­2x  ­ 2x ­ 2 30.  3x  ­ x  ­ x ­ 6 and then adds 8. The resulting 34.  11x + 8 38.  ­x  + 4x  + x ­ 5 number is 228. What number did 42.  ­9x  ­ 9x ­ 7 46.  n  ­ n + 1 Maya start with? 50.  3x + 7 54.  ­3x  ­ 2x + 10 58.  4x ­ 4 Sep 28­7:39 AM May 11­12:56 PM To multiply 2 polynomials: Multiplying Monomials • Multiply coefficients • Add exponents of like bases Examples: Remem ber th 1. 5(-7x) = you mul tiply th at whe n bases, e same you AD expone D the nt. Don 2. -5y(-7x 2) = to mul tiply co t forg et efficien ts! 3. 5x3y2(7xy4) = 4. 5xy(-9x2y3)(4x3y) = 5. 3x(-2x2y3)(-5x3y) = May 11­12:56 PM May 14­11:31 AM 5
    • Polynomials Notebook.notebook October 02, 2012 Can you put together the "old" stuff you learned yesterday with the "new" stuff from today??? Power to a Power • Distribute the outside exponent Simplify: to all inside exponents OR 1. (x - 2y) - 4(-x + y) + 2(-4x + 5y) • Rewrite what is inside the parenthesis the same # as the outside exponent and follow your rules for multiplying polynomials. 2. 2m4n(7m5n3 + 6mn2) - 6mn2(2m4n - 7m5n3) Jul 12­9:14 AM May 14­12:51 PM Examples: 1. (3xy2)3 = Homework: 2. (-x4y5)4 = p. 120 #6- 78 3. (-5a2b2c)3 = Multiples of 6! May 14­12:51 PM May 14­12:57 PM Answers to p. 120 #6­ 78 (multiples of 6) 6. 12. 18. 24. 30. 36. 42. 48. 54. 60. 66. 72. 76. Sep 28­2:49 PM May 14­12:57 PM 6
    • Polynomials Notebook.notebook October 02, 2012 Warm Up Multiplying Polynomials Aidan spent $14.95 on ingredients for cookies that will be sold at a bake sale. If each cookie will sell for $0.40, how many cookies must Aidan sell to earn a profit? May 14­1:01 PM May 14­1:01 PM May 11­12:56 PM May 14­1:10 PM May 14­1:10 PM May 14­1:11 PM 7
    • Polynomials Notebook.notebook October 02, 2012 Multiplying 2 Binomials See next page for the process and answer... May 14­1:11 PM May 14­1:12 PM May 14­1:13 PM May 14­1:13 PM A square has a side A rectangle has a length represented by x -2. Find the represented by 5x + 3 and a area of the square in terms of width represented by 2x - 7. x. Find the area and write your answer in standard form. May 14­1:14 PM May 14­1:14 PM 8
    • Polynomials Notebook.notebook October 02, 2012 A checker board is made up of 16 congruent squares. Each square has a length represented by x - 7. Find the area of 1 square in terms of x. Then find the area of the entire checker board in terms of x. May 14­1:14 PM May 14­1:14 PM Homework: p. 127 #2- 58 eoe May 14­1:15 PM May 14­1:16 PM Quiz Today CR #2 due tomorrow! May 14­1:16 PM May 11­12:57 PM 9
    • Polynomials Notebook.notebook October 02, 2012 Warm Up Factoring A hot air balloon begins ascending at the rate of 18  feet per minute.  A second hot air balloon begins  ascending 8 minutes later at a rate of 27 feet per  minute.  How many minutes will it take for the  second hot air balloon to catch the first hot air  balloon? May 11­12:56 PM May 11­12:56 PM What is a factor? The numbers that are multiplied together are called factors of the final number. May 14­1:55 PM May 15­7:52 AM Greatest Common Factor Try factoring these examples: 1. 28y2 - 4y = 2. 7x3 + 10x2 = 3. 12x3y4 - 39 x4y3 = 4. 5x + 7x2 + 9x4 = May 15­7:58 AM May 15­8:02 AM 10
    • Polynomials Notebook.notebook October 02, 2012 Factoring the Difference of 2 Perfect Squares May 15­8:02 AM May 15­8:10 AM Try factoring these examples: 1. x2 - 100 = 2. 9 - 25y2 = 3. 16x2 - 1 = 4. x6 - y10 = 5. 49x2 - 64y2 = May 15­8:12 AM May 15­8:13 AM Homework: p. 135 #28- 40 even p. 142 #1- 13 odd May 15­8:13 AM May 14­1:55 PM 11
    • Polynomials Notebook.notebook October 02, 2012 Warm Up The formula 180﴾n ­ 2﴿ is used to find the  sum of the interior angles of any  polygon.  If n represents the number of  sides the polygon has, find the total sum  of the interior angles of an octagon.   May 15­8:18 AM May 11­12:56 PM Factoring  Trinomials May 11­12:56 PM May 15­8:20 AM Heres a little trick Heres a little trick (Bigger/ Both) (Bigger/ Both) to help you figure out the signs. to help you figure out the signs. What 2 #s multiply to What 2 #s multiply to = (x )(x ) give you 12 but ADD to = (x )(x ) give you 45 but give you 7? SUBTRACT to give you 4? = (x 3)(x 4) = (x 9)(x 5) Positive. Negative. - for the letter i for Therefore both = (x + 3)(x + 4) + for the letter t for Therefore the = (x - 9)(x + 5) the word "BIGGER." signs are bigger # (9) gets the word "BOTH." The bigger # gets the positive. the - and the Both signs are the same sign as the same as the middle smaller # (5) middle sign. sign. gets the +. May 15­8:28 AM May 15­8:28 AM 12
    • Polynomials Notebook.notebook October 02, 2012 May 15­8:25 AM May 15­10:05 AM May 15­10:07 AM May 15­8:27 AM Homework: Worksheet May 15­10:10 AM May 15­8:18 AM 13
    • Polynomials Notebook.notebook October 02, 2012 Warm Up Akira scored 18 points in her last basketball game, p of which came in the 4th quarter of the game. Write an expression that can be used to find the number of points she scored in the first 3 quarters. May 15­8:18 AM May 11­12:57 PM g by  Lets start out simple...Factor the following: Factorin l pop  group ing wil  to  1. u go on up if yo lculus! tak e Ca Factoring by 2. Grouping May 11­12:57 PM May 15­10:39 AM KEY  PO INT! !! May 15­10:44 AM May 15­10:56 AM 14
    • Polynomials Notebook.notebook October 02, 2012 The Arc Method of Factoring May 15­10:57 AM May 15­11:15 AM Factor: The Arc Method of Factoring 5x2 ­ 8x + 3 Steps: 1. Take the coefficient in front of the x2 and multiply it by Wait there are no 2 #s that multiply to the last term. give me 3 and add to give me 8!!! 2. Replace the last term with this new number. (3)(1) = 3 3. Factor this new trinomial. 3+1=4 4. "Un­ arc" the factors by dividing the #s by the original # in from of the x2. 5. Reduce all fractions. What am I supposed to do???? 6. If the fraction is not divisible, move the denominator in from of your x term. We will use the Arc Method of Factoring! May 15­11:23 AM May 15­11:17 AM = 5x2 - 8x + 3 5(3) = 15 = x2 - 8x + 15 = (x - 5)(x - 3) "Un- Arcing" = (x - )(x - ) = (x - 1)(5x - 3) Multiply to check this answer!!! May 15­11:22 AM May 15­11:32 AM 15
    • Polynomials Notebook.notebook October 02, 2012 May 15­11:36 AM May 15­11:37 AM Unit 2- Quiz #3 Homework: today! Worksheet CR #2 due tomorrow May 15­10:39 AM May 11­12:58 PM Warm Up Factoring Completely The Drama Club is designing a set for an  upcoming show.  The structure is shown in the  picture.  Layer one has 1 triangle, but layer 2  consists of 3 triangles.  How many triangles will  make up the 5th layer? May 11­12:57 PM May 11­12:57 PM 16
    • Polynomials Notebook.notebook October 02, 2012 To factor x 3 + 7x2 + 12x you need to   Types of Factoring: FACTOR COMPLETELY 1.  GCF Example:  3x2 + 9x3 + 12x = 3x ﴾1 + 3x + 4x﴿ 4 2 2 Factoring completely is a three step  process: 2.  Difference between 2 perfect squares 1.  Factor a GCF from the expression, if  Example:  y2 ­ 9 = ﴾y + 3﴿﴾y ­ 3﴿ possible.  2.  Factor a Trinomial, if possible.  3.  Trinomials 3.  Factor a Difference Between Two  Example:  x2 ­ 2x ­ 3 = ﴾x ­ 3﴿﴾x + 1﴿ Squares as many times as possible.  How would you factor x 3 + 7x  + 12x??? 2 x3 + 7x  + 12x = x﴾x 2 2 + 7x + 12﴿          x﴾x + 3﴿﴾x + 4﴿   May 15­1:39 PM May 15­1:48 PM Factor each of the following COMPLETELY: Factor each of the following COMPLETELY: 1.  2x ­ 8  2 1.  18x + 39x2 ­ 15x  3 2.  n3 ­ 49n 2.  3x2 + 15x ­ 18 3.  25x2 ­ 100 May 15­1:55 PM May 15­1:55 PM Homework: Worksheet Quiz tomorrow!!! May 16­8:11 AM May 16­8:11 AM 17
    • Polynomials Notebook.notebook October 02, 2012 Warm Up Solving Quadratic What is the next number in the Equations sequence? 1, 2, 4, 8, 16, ... May 16­8:13 AM May 16­8:13 AM What is a quadratic? A quadratic equation??? May 11­12:58 PM May 16­8:17 AM It must get harder than this, right??? What do we call the highest power of a polynomial???? May 16­8:22 AM May 16­8:19 AM 18
    • Polynomials Notebook.notebook October 02, 2012 Solve the equation below for x. How many values will you end up with? Why? May 16­8:25 AM May 16­8:29 AM Heres an interesting fact... May 16­8:30 AM May 16­8:34 AM Remember the Arc Method!!! 3x2 + 13x - 10 x2 + 13x - 30 (x + 15)(x - 2) (x + )(x - ) (x +5)(3x -2) May 16­8:34 AM May 16­8:34 AM 19
    • Polynomials Notebook.notebook October 02, 2012 What do I do if the quadratic doesnt have 4 parts? May 16­8:33 AM May 16­8:33 AM May 16­8:33 AM May 16­8:33 AM Heres a super dooper hard one! Homework: Worksheet Review tomorrow! Test in 2 days =) May 16­8:33 AM May 16­11:45 AM 20
    • Polynomials Notebook.notebook October 02, 2012 May 16­11:45 AM May 11­12:58 PM 21