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Slide 1 / 127Polynomials

Slide 2 / 127Table of Contents········Factors and GCFFactoring out GCF'sFactoring Polynomial w/ a 1Factoring Using Special PatternsFactoring Trinomial a#1Factoring 4 Term PolynomialsMixed FactoringSolving Equations by Factoring

Slide 4 / 127Factors of 10FactorsUniqueto 10Factors of 15FactorsUniqueto 15Factors 10 and 15have in commonNumberBank111212313456789101415What is the greatest common factor (GCF) of 10 and 15?1617181920

Slide 5 / 127Factors of 12FactorsUniqueto 12Factors of 18FactorsUniqueto 18Factors 12 and 18have in commonNumberBank111212313456789101415What is the greatest common factor (GCF) of 12 and 18?1617181920

Slide 6 / 1271What is the GCF of 12 and 15?

Slide 7 / 1272What is the GCF of 24 and 48?

Slide 8 / 1273What is the GCF of 72 and 54?

Slide 9 / 1274What is the GCF of 70 and 99?

Slide 10 / 1275What is the GCF of 28, 56 and 42?

Slide 11 / 127Variables also have a GCF.The GCF of variables is the variable(s) that is in each term raised tothe lowest exponent given.Example: Find the GCFandandandandandand

Slide 12 / 1276What is the GCF ofABCDand?

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Slide 17 / 127The first step in factoring is to determine its greatest monomial factor. Ifthere is a greatest monomial factor other than 1, use the distributiveproperty to rewrite the given polynomial as the product of this greatestmonomial factor and a polynomial.Example 1a)Factor each polynomial.6x4 - 15x3 3x2Find the GCFGCF: 3x2Reduce each term ofthe polynomial dividingby the GCF3x2 (2x2 - 5x 1)

Slide 18 / 127The first step in factoring is to determine its greatest monomial factor. Ifthere is a greatest monomial factor other than 1, use the distributiveproperty to rewrite the given polynomial as the product of this greatestmonomial factor and a polynomial.Example 1Factor each polynomial.b) 4m3n - 7m2n2Find the GCFGCF: m2nReduce each term ofthe polynomial dividingby the GCFm2n(4n - 7n)

Slide 19 / 127Sometimes the distributive property can be used to factor apolynomial that is not in simplest form but has a common binomialfactor.Example 2a)Factor each polynomial.y(y - 3) 7(y - 3)Find the GCFGCF: y - 3Reduce each term ofthe polynomial dividingby the GCF(y - 3)(y 7)

Slide 20 / 127Sometimes the distributive property can be used to factor apolynomial that is not in simplest form but has a common binomialfactor.Example 2b)Factor each polynomial.Find the GCFGCF:Reduce each term ofthe polynomial dividingby the GCF

Slide 21 / 127In working with common binomial factors, look for factors that areopposites of each other.For example:(x - y) - (y - x) becausex - y x (-y) -y x -(y - x)

Slide 22 / 12710 True or False: y - 7 -1( 7 y)TrueFalse

Slide 23 / 12711 True or False: 8 - d -1( d 8)TrueFalse

Slide 24 / 12712 True or False: 8c - h -1( -8c h)TrueFalse

Slide 25 / 12713 True or False: -a - b -1( a b)TrueFalse

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Slide 28 / 12714 If possible, FactorABCD Already Simplified

Slide 29 / 12715 If possible, FactorABCD Already Simplified

Slide 30 / 12716 If possible, FactorABCD Already Simplified

Slide 31 / 12717 If possible, FactorABCD Already Simplified

Slide 32 / 12718 If possible, FactorABCD Already Simplified

Slide 33 / 12719 If possible, FactorABCD Already Simplified

Slide 35 / 127A polynomial that can be simplified to the formax bx c, where a 0, is called a quadratic polynomial.QuaCoedrar nstaattenticrtemte.rmrm.Lin

A quadratic polynomial in which b 0 and c 0 is called aquadratic trinomial. If only b 0 or c 0 it is called a quadraticbinomial. If both b 0 and c 0 it is a quadratic monomial.Slide 36 / 127Examples: Choose all of the description that lMonomial

Slide 37 / 12720 Choose all of the descriptions that apply onomial

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Slide 39 / 12722 Choose all of the descriptions that apply onomial

Slide 40 / 12723 Choose all of the descriptions that apply onomial

Slide 41 / 127Simplify.1)2)3)4)(x 2)(x 3) (x - 4)(x - 1) (x 1)(x - 5) (x 6)(x - 2) AnswerBankx2 - 5x 4x2 - 4x - 5x2 5x 6x2 4x - 12RECALL What did we do? Look for a pattern!!Slide each polynomialfrom the circle to thecorrect expression.

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Slide 44 / 127Examples:

Slide 45 / 12724 The factors of 12 will have what kind of signs given thefollowing equation?ABoth positiveBBoth NegativeCBigger factor positive, the other negativeDThe bigger factor negative, the other positive

Slide 46 / 12725 The factors of 12 will have what kind of signs given thefollowing equation?ABoth positiveBBoth negativeCBigger factor positive, the other negativeDThe bigger factor negative, the other positive

Slide 47 / 12726 FactorA(x 12)(x 1)B(x 6)(x 2)C(x 4)(x 3)D(x - 12)(x - 1)E(x - 6)(x - 1)F(x - 4)(x - 3)

Slide 48 / 12727 FactorA(x 12)(x 1)B(x 6)(x 2)C(x 4)(x 3)D(x - 12)(x - 1)E(x - 6)(x - 1)F(x - 4)(x - 3)

Slide 49 / 12728 FactorA(x 12)(x 1)B(x 6)(x 2)C(x 4)(x 3)D(x - 12)(x - 1)E(x - 6)(x - 1)F(x - 4)(x - 3)

Slide 50 / 12729 FactorA(x 12)(x 1)B(x 6)(x 2)C(x 4)(x 3)D(x - 12)(x - 1)E(x - 6)(x - 1)F(x - 4)(x - 3)

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Slide 53 / 127Examples

Slide 54 / 12730 The factors of -12 will have what kind of signs given thefollowing equation?ABoth positiveBBoth negativeCBigger factor positive, the other negativeDThe bigger factor negative, the other positive

Slide 55 / 12731 The factors of -12 will have what kind of signs given thefollowing equation?ABoth positiveBBoth negativeCBigger factor positive, the other negativeDThe bigger factor negative, the other positive

Slide 56 / 12732 FactorA(x 12)(x - 1)B(x 6)(x - 2)C(x 4)(x - 3)D(x - 12)(x 1)E(x - 6)(x 1)F(x - 4)(x 3)

Slide 57 / 12733 FactorA(x 12)(x - 1)B(x 6)(x - 2)C(x 4)(x - 3)D(x - 12)(x 1)E(x - 6)(x 1)F(x - 4)(x 3)

Slide 58 / 12734 FactorA(x 12)(x - 1)B(x 6)(x - 2)C(x 4)(x - 3)D(x - 12)(x 1)E(x - 6)(x 1)F(x - 4)(x 3)

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Slide 60 / 127Mixed Factoring

Slide 61 / 12736 Factor the followingA(x - 2)(x - 4)B(x 2)(x 4)C(x - 2)(x 4)D(x 2)(x - 4)

Slide 62 / 12737 Factor the followingA(x - 3)(x - 5)B(x 3)(x 5)C(x - 3)(x 5)D(x 3)(x - 5)

Slide 63 / 12738 Factor the followingA(x - 3)(x - 4)B(x 3)(x 4)C(x 2)(x 6)D(x 1)(x 12)

Slide 64 / 12739 Factor the followingA(x - 2)(x - 5)B(x 2)(x 5)C(x - 2)(x 5)D(x 2)(x - 5)

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Slide 68 / 127Factor:Factor outSTEP1STEP2STEP3STEP4

Slide 69 / 12740 Factor completely:ABCD

Slide 70 / 12741 Factor completely:ABCD

Slide 71 / 12742 Factor completely:ABCD

Slide 72 / 12743 Factor completely:ABCD

Slide 73 / 12744 Factor completely:ABCD

Slide 75 / 127When we were multiplying polynomials we had specialpatterns.Square of SumsDifference of SumsProduct of a Sum and a DifferenceIf we learn to recognize these squares and products we can usethem to help us factor.

Slide 76 / 127Perfect Square TrinomialsThe Square of a Sum and the Square of a difference have productsthat are called Perfect Square Trinomials.How to Recognize a Perfect Square Trinomial:Recall:Observe the trinomial. The first term is a perfect square.The second term is 2 times square root ofthe first term times the square root of thethird. The sign is plus/minus.The third term is a perfect square.

Slide 77 / 127Examples of Perfect Square Trinomials

Slide 78 / 127Is the trinomial a perfect square?Drag the Perfect SquareTrinomials into the Box.Only Perfect Square Trinomialswill remain visible.

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Slide 80 / 12745 FactorABCD Not a perfect Square Trinomial

Slide 81 / 12746 FactorABCD Not a perfect Square Trinomial

Slide 82 / 12747 FactorABCD Not a perfect Square Trinomial

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Slide 84 / 127Examples of Difference of Squares

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Slide 87 / 12748 FactorABCDNot a Difference of Squares

Slide 88 / 12750 FactorABCDNot a Difference of Squares

Slide 89 / 12751 FactorABCDNot a Difference of Squares

Slide 90 / 12752 Factor using Difference of Squares:ABCDNot a Difference of Squares

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Slide 93 / 127How to factor a trinomial of the form ax² bx c.Example:Factor 2d² 15d 18Find the product of a and c :2 18 36Now find two integers whose product is 36 and whose sum isequal to “b” or 15. Factors of 36Sum 15?1 36 371, 362 18 202, 183 12 153, 12Now substitute 12 3 into the equation for 15.2d² (12 3)d 18Distribute2d² 12d 3d 18Group and factor GCF2d(d 6) 3(d 6)Factor common binomial(d 6)(2d 3)Remember to check using FOIL!

Factor.15x² - 13x 2Slide 94 / 127a 15 and c 2, but b -13Since both a and c are positive, and b is negative we needto find two negative factors of 30 that add up to -13Factors of 30-1, -30-2, -15-3, -10-5, -6Sum -13?-1 -30 -31-2 -15 -17-3 -10 -13-5 -6 -11

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Slide 96 / 127Factor6y² - 13y - 5

Slide 97 / 127A polynomial that cannot be written as a product of twopolynomials is called a prime polynomial.

Slide 98 / 12753 FactorABCDPrime Polynomial

Slide 99 / 12754 FactorABCDPrime Polynomial

Slide 100 / 12755 FactorABCDPrime Polynomial

Polynomials with four terms like ab - 4b 6a - 24, canSlide 102 / 127be factored by grouping terms of the polynomials.Example 1:ab - 4b 6a - 24(ab - 4b) (6a - 24)b(a - 4) 6(a - 4)(a - 4) (b 6)Group terms into binomials that can befactored using the distributive propertyFactor the GCFNotice that a - 4 is a common binomialfactor and factor!

Slide 103 / 127Example 2:6xy 8x - 21y - 28(6xy 8x) (-21y - 28)2x(3y 4) (-7)(3y 4)(3y 4) (2x - 7)GroupFactor GCFFactor common binomial

Slide 104 / 127You must be able to recognize additive inverses!!!(3 - a and a - 3 are additive inverses because their sum is equal to zero.)Remember 3 - a -1(a - 3).Example 3:15x - 3xy 4y - 20(15x - 3xy) (4y - 20)3x(5 - y) 4(y - 5)3x(-1)(-5 y) 4(y - 5)-3x(y - 5) 4(y - 5)(y - 5) (-3x 4)GroupFactor GCFNotice additive inversesSimplifyFactor common binomialRemember to check each problem by using FOIL.

Slide 105 / 12756 Factor 15ab - 3a 10b - 2A(5b - 1)(3a 2)B(5b 1)(3a 2)C(5b - 1)(3a - 2)D(5b 1)(3a - 1)

Slide 106 / 12757 Factor 10m2n - 25mn 6m - 15A(2m-5)(5mn-3)B(2m-5)(5mn 3)C(2m 5)(5mn-3)D(2m 5)(5mn 3)

Slide 107 / 12758 Factor 20ab - 35b - 63 36aA(4a - 7)(5b - 9)B(4a - 7)(5b 9)C(4a 7)(5b - 9)D(4a 7)(5b 9)

Slide 108 / 12759 Factor a2 - ab 7b - 7aA(a - b)(a - 7)B(a - b)(a 7)C(a b)(a - 7)D(a b)(a 7)

Slide 110 / 127Summary of FactoringFactor the PolynomialFactor out GCF2 TermsDifferenceof Squares4 Terms3 TermsPerfect SquareTrinomialFactor theTrinomiala 1Group and Factorout GCF. Look for aCommon Binomiala 1Check each factor to see if it can be factored again.If a polynomial cannot be factored, then it is called prime.

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Slide 112 / 12760 Factor completely:ABCD

Slide 113 / 12761 Factor completelyABCD prime polynomial

Slide 114 / 12762 FactorABCDprime polynomial

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Slide 116 / 12764 FactorABCDPrime Polynomial

Slide 118 / 127Given the following equation, what conclusion(s) can bedrawn?ab 0Since the product is 0, one of the factors, a or b, must be 0.This is known as the Zero Product Property.

Slide 119 / 127Given the following equation, what conclusion(s) can be drawn?(x - 4)(x 3) 0Since the product is 0, one of the factors must be 0.Therefore, either x - 4 0 or x 3 0.x - 4 0 4 4x 4ororx 3 0-3 -3x -3Therefore, our solution set is {-3, 4}. To verify the results, substitute eachsolution back into the original equation.To check x 4: (x - 4)(x 3) 0To check x -3: (x - 4)(x 3) 0(-3 - 4)(-3 3) 0(-7)(0) 00 0(4 - 4)(4 3) 0(0)(7) 00 0

Slide 120 / 127What if you were given the following equation?How would you solve it?We can use the Zero Product Property to solve it.How can we turn this polynomial into a multiplication problem? Factor it!Factoring yields:(x - 6)(x 4) 0By the Zero Product Property:x-6 0orx 4 0After solving each equation, we arrive at our solution:{-4, 6}

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Slide 124 / 12765 Choose all of the solutions to:ABCDEF

Slide 125 / 12766 Choose all of the solutions to:ABCDEF

Slide 126 / 12767 Choose all of the solutions to:ABCDEF

Slide 127 / 12768 A ball is thrown with its height at any time given byWhen does the ball hit the ground?A-1 secondsB0 secondsC9 secondsD10 seconds

May 13, 2009 · · Factoring out GCF's · Factoring 4 Term Polynomials · Factoring Polynomial w/ a 1 · Factoring Using Special Patterns · Factoring Trinomial a#1 · Mixed Factoring · Solving Equations by