Slide 1 / 128. Polynomials

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1 Slide 1 / 128 Polynomials

2 Slide 2 / 128 Table of Contents Factors and GCF Factoring out GCF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring 4 Term Polynomials Mixed Factoring Solving Equations by Factoring

3 Slide 3 / 128 Factors and Greatest Common Factors Return to Table of Contents

4 Slide 4 / 128 Factors of 10 Factors of 15 Number Bank Factors Unique to 10 Factors 10 and 15 have in common Factors Unique to What is the greatest common factor (GCF) of 10 and 15?

5 Slide 5 / 128 Factors of 12 Factors of 18 Number Bank Factors Unique to 12 Factors 12 and 18 have in common Factors Unique to What is the greatest common factor (GCF) of 12 and 18?

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

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

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

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

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

11 Slide 11 / 128 Variables also have a GCF. The GCF of variables is the variable(s) that is in each term raised to the lowest exponent given. Example: Find the GCF and and and and and and

12 Slide 12 / What is the GCF of and? A B C D

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16 Slide 16 / 128 Factoring out GCFs Return to Table of Contents

17 Slide 17 / 128 The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial. Example 1 Factor each polynomial. a) 6x 4-15x 3 + 3x 2 Find the GCF 3x 2 6x 4 15x 3 3x 2 GCF: 3x 2 3x 2 3x 2 3x 2 Reduce each term of the polynomial dividing by the GCF 3x 2 (2x 2-5x + 1)

18 Slide 18 / 128 The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial. Example 1 Factor each polynomial. b) 4m 3 n - 7m 2 n 2 Find the GCF GCF: m 2 n Reduce each term of the polynomial dividing by the GCF m 2 n(4n - 7n)

19 Slide 19 / 128 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Example 2 Factor each polynomial. a) y(y - 3) + 7(y - 3) Find the GCF GCF: y - 3 (y - 3) ( y(y - 3) (y - 3) + 7(y - 3) (y - 3) ( Reduce each term of the polynomial dividing by the GCF (y - 3)(y + 7)

20 Slide 20 / 128 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Example 2 Factor each polynomial. b) Find the GCF GCF: Reduce each term of the polynomial dividing by the GCF

21 Slide 21 / 128 In working with common binomial factors, look for factors that are opposites of each other. For example: (x - y) = - (y - x) because x - y = x + (-y) = -y + x = -(y - x)

22 Slide 22 / True or False: y - 7 = -1( 7 + y) True False

23 Slide 23 / True or False: 8 - d = -1( d + 8) True False

24 Slide 24 / True or False: 8c - h = -1( -8c + h) True False

25 Slide 25 / True or False: -a - b = -1( a + b) True False

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28 Slide 28 / If possible, Factor A B C D Already Simplified

29 Slide 29 / If possible, Factor A B C D Already Simplified

30 Slide 30 / If possible, Factor A B C D Already Simplified

31 Slide 31 / If possible, Factor A B C D Already Simplified

32 Slide 32 / If possible, Factor A B C D Already Simplified

33 Slide 33 / If possible, Factor A B C D Already Simplified

34 Slide 34 / 128 Factoring Trinomials: x 2 + bx + c Return to Table of Contents

35 Slide 35 / 128 A polynomial that can be simplified to the form ax + bx + c, where a 0, is called a quadratic polynomial. Quadratic term. Linear term. Constant term.

36 Slide 36 / 128 A quadratic polynomial in which b 0 and c 0 is called a quadratic trinomial. If only b=0 or c=0 it is called a quadratic binomial. If both b=0 and c=0 it is a quadratic monomial. Examples: Choose all of the description that apply. Cubic Quadratic Linear Constant Trinomial Binomial Monomial

Slide 37 / 128 20 Choose all of the descriptions that apply to:

37 Slide 37 / Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial

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39 Slide 39 / Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial

40 Slide 40 / Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial

41 Slide 41 / 128 Simplify. 1) (x + 2)(x + 3) = 2) (x - 4)(x - 1) = 3) (x + 1)(x - 5) = 4) (x + 6)(x - 2) = Answer Bank x 2-5x + 4 x 2 + 5x + 6 x 2-4x - 5 x 2 + 4x - 12 Slide each polynomial from the circle to the correct expression. RECALL What did we do?? Look for a pattern!!

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44 Slide 44 / 128 Examples: (x - 8)(x - 1)

45 Slide 45 / The factors of 12 will have what kind of signs given the following equation? A B C D Both positive Both Negative Bigger factor positive, the other negative The bigger factor negative, the other positive

46 Slide 46 / The factors of 12 will have what kind of signs given the following equation? A B C D Both positive Both negative Bigger factor positive, the other negative The bigger factor negative, the other positive

47 Slide 47 / Factor A (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)

48 Slide 48 / Factor A (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)

49 Slide 49 / Factor A (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)

50 Slide 50 / Factor A (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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53 Examples Slide 53 / 128

54 Slide 54 / The factors of -12 will have what kind of signs given the following equation? A B C D Both positive Both negative Bigger factor positive, the other negative The bigger factor negative, the other positive

55 Slide 55 / The factors of -12 will have what kind of signs given the following equation? A B C D Both positive Both negative Bigger factor positive, the other negative The bigger factor negative, the other positive

56 Slide 56 / Factor x 2-4x - 12 A (x + 12)(x - 1) B (x + 6)(x - 2) C (x + 4)(x - 3) D (x - 12)(x + 1) E (x - 6)(x + 2) F (x - 4)(x + 3)

57 Slide 57 / Factor A (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)

58 Slide 58 / Factor A (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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60 Slide 60 / 128 Mixed Practice

61 Slide 61 / Factor the following A (x - 2)(x - 4) B (x + 2)(x + 4) C (x - 2)(x +4) D (x + 2)(x - 4)

62 Slide 62 / Factor the following A (x - 3)(x - 5) B (x + 3)(x + 5) C (x - 3)(x +5) D (x + 3)(x - 5)

63 Slide 63 / Factor the following A (x - 3)(x - 4) B (x + 3)(x + 4) C (x +2)(x +6) D (x + 1)(x+12)

64 Slide 64 / Factor the following A (x - 2)(x - 5) B (x + 2)(x + 5) C (x - 2)(x +5) D (x + 2)(x - 5)

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68 Slide 68 / 128 Factor: Factor out STEP 1 STEP 2 STEP 3 STEP 4

69 Slide 69 / Factor completely: A B C D

70 Slide 70 / Factor completely: A B C D

71 Slide 71 / Factor completely: A B C D

72 Slide 72 / Factor completely: A B C D

73 Slide 73 / Factor completely: A B C D

74 Slide 74 / 128 Factoring Using Special Patterns Return to Table of Contents

75 Slide 75 / 128 When we were multiplying polynomials we had special patterns. Square of Sums Difference of Sums Product of a Sum and a Difference If we learn to recognize these squares and products we can use them to help us factor.

76 Slide 76 / 128 Perfect Square Trinomials The Square of a Sum and the Square of a difference have products that 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 of the first term times the square root of the third. The sign is plus/minus. The third term is a perfect square.

77 Slide 77 / 128 Examples of Perfect Square Trinomials

78 Slide 78 / 128 Is the trinomial a perfect square? Drag the Perfect Square Trinomials into the Box. Only Perfect Square Trinomials will remain visible.

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80 Slide 80 / Factor A B C D Not a perfect Square Trinomial

81 Slide 81 / Factor A B C D Not a perfect Square Trinomial

82 Slide 82 / Factor A B C D Not a perfect Square Trinomial

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84 Slide 84 / 128 Examples of Difference of Squares

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87 Slide 87 / Factor A B C D Not a Difference of Squares

88 Slide 88 / Factor A B C D Not a Difference of Squares

89 Slide 89 / Factor A B C D Not a Difference of Squares

Slide 90 / 128 51 Factor using Difference of

90 Slide 90 / Factor using Difference of Squares: A B C D Not a Difference of Squares

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92 Slide 92 / 128 Factoring Trinomials: ax 2 + bx + c Return to Table of Contents

93 Slide 93 / 128 How to factor a trinomial of the form ax² + bx + c. Example: Factor 2d² + 15d + 18 Find the product of a and c : 2 18 = 36 Now find two integers whose product is 36 and whose sum is equal to b or 15. Factors of 36 Sum = 15? 1, 36 2, 18 3, = = = 15 Now substitute into the equation for 15. 2d² + (12 + 3)d + 18 Distribute 2d² + 12d + 3d + 18 Group and factor GCF 2d(d + 6) + 3(d + 6) Factor common binomial (d + 6)(2d + 3) Remember to check using FOIL!

94 Slide 94 / 128 Factor. 15x² - 13x + 2 a = 15 and c = 2, but b = -13 Since both a and c are positive, and b is negative we need to find two negative factors of 30 that add up to -13 Factors of 30 Sum = -13? -1, -30-2, -15-3, -10-5, = = = = -11

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96 Factor 6y² - 13y - 5 Slide 96 / 128

97 Slide 97 / 128 A polynomial that cannot be written as a product of two polynomials is called a prime polynomial.

98 Slide 98 / Factor A B C D Prime Polynomial

99 Slide 99 / Factor A B C D Prime Polynomial

100 Slide 100 / Factor A B C D Prime Polynomial

101 Slide 101 / 128 Factoring 4 Term Polynomials Return to Table of Contents

102 Slide 102 / 128 Polynomials with four terms like ab - 4b + 6a - 24, can be factored by grouping terms of the polynomials. Example 1: ab - 4b + 6a - 24 (ab - 4b) + (6a - 24) Group terms into binomials that can be factored using the distributive proper b(a - 4) + 6(a - 4) Factor the GCF (a - 4) (b + 6) Notice that a - 4 is a common binomial factor and factor!

103 Slide 103 / 128 Example 2: 6xy + 8x - 21y - 28 (6xy + 8x) + (-21y - 28) Group 2x(3y + 4) + (-7)(3y + 4) Factor GCF (3y +4) (2x - 7) Factor common binomial

104 Slide 104 / 128 You must be able to recognize additive inverses!!! (3 - a and a - 3 are additive inverses because their sum is equal to Remember 3 - a = -1(a - 3). Example 3: 15x - 3xy + 4y - 20 (15x - 3xy) + (4y - 20) Group 3x(5 - y) + 4(y - 5) Factor GCF 3x(-1)(-5 + y) + 4(y - 5) Notice additive inverses -3x(y - 5) + 4(y - 5) Simplify (y - 5) (-3x + 4) Factor common binomial Remember to check each problem by using FOIL.

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

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

107 Slide 107 / Factor 20ab - 35b a A (4a - 7)(5b - 9) B (4a - 7)(5b + 9) C (4a + 7)(5b - 9) D (4a + 7)(5b + 9)

108 Slide 108 / Factor a 2 - ab + 7b - 7a A (a - b)(a - 7) B (a - b)(a + 7) C (a + b)(a - 7) D (a + b)(a + 7)

109 Slide 109 / 128 Mixed Factoring Return to Table of Contents

110 Slide 110 / 128 Summary of Factoring Factor the Polynomial 2 Terms Difference of Squares Perfect Square Trinomial Factor out GCF 3 Terms Factor the Trinomial 4 Terms Group and Factor out GCF. Look for a Common Binomial a = 1 a = 1 Check 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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112 Slide 112 / Factor completely: A B C D

113 Slide 113 / Factor completely A B C D prime polynomial

114 Slide 114 / Factor A B C D prime polynomial

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116 Slide 116 / Factor A B C D Prime Polynomial

117 Slide 117 / 128 Solving Equations by Factoring Return to Table of Contents

118 Slide 118 / 128 Given the following equation, what conclusion(s) can be drawn? ab = 0 Since the product is 0, one of the factors, a or b, must be 0. This is known as the Zero Product Property.

119 Slide 119 / 128 Given the following equation, what conclusion(s) can be dr (x - 4)(x + 3) = 0 Since the product is 0, one of the factors must be 0. Therefore, either x - 4 = 0 or x + 3 = 0. x - 4 = 0 or x + 3 = x = 4 or x = -3 Therefore, our solution set is {-3, 4}. To verify the results, substitute solution back into the original equation. To check x = -3: (x - 4)(x + 3) = 0 To check x = 4: (x - 4)(x + 3) = 0 (-3-4)(-3 + 3) = 0 (-7)(0) = 0 0 = 0 (4-4)(4 + 3) = 0 (0)(7) = 0 0 = 0

120 Slide 120 / 128 What 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? Fac Factoring yields: (x - 6)(x + 4) = 0 By the Zero Product Property: x - 6 = 0 or x + 4 = 0 After solving each equation, we arrive at our solution: {-4, 6}

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124 Slide 124 / Choose all of the solutions to: A B C D E F

125 Slide 125 / Choose all of the solutions to: A B C D E F

126 Slide 126 / Choose all of the solutions to: A B C D E F

127 Slide 127 / A ball is thrown with its height at any time given by A B C D When does the ball hit the ground? -1 seconds 0 seconds 9 seconds 10 seconds

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Polynomials. Factors and Greatest Common Factors. Slide 1 / 128. Slide 2 / 128. Slide 3 / 128. Table of Contents

Polynomials. Factors and Greatest Common Factors. Slide 1 / 128. Slide 2 / 128. Slide 3 / 128. Table of Contents Slide 1 / 128 Polynomials Table of ontents Slide 2 / 128 Factors and GF Factoring out GF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring