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# AA Section 11-5

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The Factor Theorem

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• ### AA Section 11-5

1. 1. Section 11-5 The Factor Theorem Sunday, March 15, 2009
2. 2. In-Class Activity 1. What were the x-intercepts for number 1? What was the factored form of the polynomial? 2. What were the x-intercepts in number 2? (x - 1)(x + 1)(x - 3)(x + 4) Sunday, March 15, 2009
3. 3. In-Class Activity 1. What were the x-intercepts for number 1? What was the factored form of the polynomial? x = -4, 0, 3 2. What were the x-intercepts in number 2? (x - 1)(x + 1)(x - 3)(x + 4) Sunday, March 15, 2009
4. 4. In-Class Activity 1. What were the x-intercepts for number 1? What was the factored form of the polynomial? x = -4, 0, 3 x(x - 3)(x + 4) 2. What were the x-intercepts in number 2? (x - 1)(x + 1)(x - 3)(x + 4) Sunday, March 15, 2009
5. 5. In-Class Activity 1. What were the x-intercepts for number 1? What was the factored form of the polynomial? x = -4, 0, 3 x(x - 3)(x + 4) ...interesting. 2. What were the x-intercepts in number 2? (x - 1)(x + 1)(x - 3)(x + 4) Sunday, March 15, 2009
6. 6. In-Class Activity 1. What were the x-intercepts for number 1? What was the factored form of the polynomial? x = -4, 0, 3 x(x - 3)(x + 4) ...interesting. 2. What were the x-intercepts in number 2? (x - 1)(x + 1)(x - 3)(x + 4) x = 1, -1, 3, -4 Sunday, March 15, 2009
7. 7. In-Class Activity 1. What were the x-intercepts for number 1? What was the factored form of the polynomial? x = -4, 0, 3 x(x - 3)(x + 4) ...interesting. 2. What were the x-intercepts in number 2? (x - 1)(x + 1)(x - 3)(x + 4) x = 1, -1, 3, -4 Hmm... Sunday, March 15, 2009
8. 8. In-Class Activity 1. What were the x-intercepts for number 1? What was the factored form of the polynomial? x = -4, 0, 3 x(x - 3)(x + 4) ...interesting. 2. What were the x-intercepts in number 2? (x - 1)(x + 1)(x - 3)(x + 4) x = 1, -1, 3, -4 Hmm... What can we say about what’s happening here? Sunday, March 15, 2009
9. 9. Zero-Product Theorem Sunday, March 15, 2009
10. 10. Zero-Product Theorem For all a and b, ab = 0 IFF a = 0 or b = 0 Sunday, March 15, 2009
11. 11. Zero-Product Theorem For all a and b, ab = 0 IFF a = 0 or b = 0 This means that if we multiply two numbers together and the product is zero, at least one of the numbers must be zero! Sunday, March 15, 2009
12. 12. Example 1 a. Write a polynomial to represent the volume of the box. x x 20 in. 30 in. Sunday, March 15, 2009
13. 13. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 in. 30 in. Sunday, March 15, 2009
14. 14. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 in. Width = 30 in. Sunday, March 15, 2009
15. 15. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 in. Width = Height = 30 in. Sunday, March 15, 2009
16. 16. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = Height = 30 in. Sunday, March 15, 2009
17. 17. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = 30 in. Sunday, March 15, 2009
18. 18. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. Sunday, March 15, 2009
19. 19. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = Sunday, March 15, 2009
20. 20. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = (30 - 2x) Sunday, March 15, 2009
21. 21. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = (30 - 2x)(20 - 2x) Sunday, March 15, 2009
22. 22. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = (30 - 2x)(20 - 2x)(x) Sunday, March 15, 2009
23. 23. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = (30 - 2x)(20 - 2x)(x) = Sunday, March 15, 2009
24. 24. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = (30 - 2x)(20 - 2x)(x) = (600 - 100x + 4x2)(x) Sunday, March 15, 2009
25. 25. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = (30 - 2x)(20 - 2x)(x) = (600 - 100x + 4x2)(x) = 4x3 - 100x2 + 600x Sunday, March 15, 2009
26. 26. Example 1 a. Write a polynomial to represent the volume of the box. x x Length = 20 - 2x 20 in. Width = 30 - 2x Height = x 30 in. V(x) = (30 - 2x)(20 - 2x)(x) = (600 - 100x + 4x2)(x) = 4x3 - 100x2 + 600x in3 Sunday, March 15, 2009
27. 27. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
28. 28. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
29. 29. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
30. 30. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
31. 31. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
32. 32. Sunday, March 15, 2009
33. 33. Sunday, March 15, 2009
34. 34. Sunday, March 15, 2009
35. 35. Sunday, March 15, 2009
36. 36. Sunday, March 15, 2009
37. 37. x = 0, 10, 15 Sunday, March 15, 2009
38. 38. Question: If there are two numbers that are being multiplied to get a product of 0, what can we say about at least one of the numbers? Sunday, March 15, 2009
39. 39. Factor Theorem x - r is a factor of a polynomial P(x) IFF P(r) = 0 Sunday, March 15, 2009
40. 40. Factor Theorem x - r is a factor of a polynomial P(x) IFF P(r) = 0 This means that if we have a polynomial in standard form (equal to 0), we can take each factor and set it equal to 0 to ﬁnd the zeros! Sunday, March 15, 2009
41. 41. Factor Theorem x - r is a factor of a polynomial P(x) IFF P(r) = 0 This means that if we have a polynomial in standard form (equal to 0), we can take each factor and set it equal to 0 to ﬁnd the zeros! This means a lot to us! Sunday, March 15, 2009
42. 42. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! Sunday, March 15, 2009
43. 43. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0= Sunday, March 15, 2009
44. 44. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x Sunday, March 15, 2009
45. 45. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 Sunday, March 15, 2009
46. 46. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x Sunday, March 15, 2009
47. 47. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) Sunday, March 15, 2009
48. 48. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) (-6)(-5) = 30 Sunday, March 15, 2009
49. 49. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) (-6)(-5) = 30 -6 - 5 = -11 Sunday, March 15, 2009
50. 50. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (-6)(-5) = 30 -6 - 5 = -11 Sunday, March 15, 2009
51. 51. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6) (-6)(-5) = 30 -6 - 5 = -11 Sunday, March 15, 2009
52. 52. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 -6 - 5 = -11 Sunday, March 15, 2009
53. 53. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 Sunday, March 15, 2009
54. 54. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 3x = 0 Sunday, March 15, 2009
55. 55. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 3x = 0 x-6=0 Sunday, March 15, 2009
56. 56. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 3x = 0 x-6=0 x-5=0 Sunday, March 15, 2009
57. 57. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 3x = 0 x-6=0 x-5=0 x=0 Sunday, March 15, 2009
58. 58. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 3x = 0 x-6=0 x-5=0 x=0 x=6 Sunday, March 15, 2009
59. 59. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 3x = 0 x-6=0 x-5=0 x=0 x=6 x=5 Sunday, March 15, 2009
60. 60. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! 0 = 3x(x2 - 11x + 30) = 3x (x - 6)(x - 5) (-6)(-5) = 30 Set each factor equal to 0. -6 - 5 = -11 3x = 0 x-6=0 x-5=0 x=0 x=6 x=5 Check your answers to see if they all work. Sunday, March 15, 2009
61. 61. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x Sunday, March 15, 2009
62. 62. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x Sunday, March 15, 2009
63. 63. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) Sunday, March 15, 2009
64. 64. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) 0 = 4x(x - 15)(x - 10) Sunday, March 15, 2009
65. 65. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) 0 = 4x(x - 15)(x - 10) 0 = 4x Sunday, March 15, 2009
66. 66. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) 0 = 4x(x - 15)(x - 10) 0 = 4x 0 = x - 15 Sunday, March 15, 2009
67. 67. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) 0 = 4x(x - 15)(x - 10) 0 = 4x 0 = x - 15 0 = x - 10 Sunday, March 15, 2009
68. 68. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) 0 = 4x(x - 15)(x - 10) 0 = 4x 0 = x - 15 0 = x - 10 x=0 Sunday, March 15, 2009
69. 69. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) 0 = 4x(x - 15)(x - 10) 0 = 4x 0 = x - 15 0 = x - 10 x=0 x = 15 Sunday, March 15, 2009
70. 70. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x 0 = 4x(x2 - 25x + 150) 0 = 4x(x - 15)(x - 10) 0 = 4x 0 = x - 15 0 = x - 10 x=0 x = 15 x = 10 Sunday, March 15, 2009
71. 71. Another question: Why do we call these “zeros?” Sunday, March 15, 2009
72. 72. Another question: Why do we call these “zeros?” It’s where y is equal to zero. Sunday, March 15, 2009
73. 73. Yet another question: What other names do we use for zeros? Sunday, March 15, 2009
74. 74. Yet another question: What other names do we use for zeros? Solutions, x-intercepts, roots Sunday, March 15, 2009
75. 75. Example 3 Find P(x), which has zeros of -2, 0, and 2. Sunday, March 15, 2009
76. 76. Example 3 Find P(x), which has zeros of -2, 0, and 2. Well, if we know the zeros, we know the factors! Sunday, March 15, 2009
77. 77. Example 3 Find P(x), which has zeros of -2, 0, and 2. Well, if we know the zeros, we know the factors! P(x) = x(x - 2)(x + 2) Sunday, March 15, 2009
78. 78. Example 3 Find P(x), which has zeros of -2, 0, and 2. Well, if we know the zeros, we know the factors! P(x) = x(x - 2)(x + 2) = kx(x2 + 2x - 2x - 4) Sunday, March 15, 2009
79. 79. Example 3 Find P(x), which has zeros of -2, 0, and 2. Well, if we know the zeros, we know the factors! P(x) = x(x - 2)(x + 2) = kx(x2 + 2x - 2x - 4) = kx3 - 4kx Sunday, March 15, 2009
80. 80. Example 3 Find P(x), which has zeros of -2, 0, and 2. Well, if we know the zeros, we know the factors! P(x) = x(x - 2)(x + 2) = kx(x2 + 2x - 2x - 4) = kx3 - 4kx k is a constant Sunday, March 15, 2009
81. 81. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. Sunday, March 15, 2009
82. 82. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 0 = x2(3x2 - 28x - 20) Sunday, March 15, 2009
83. 83. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) Sunday, March 15, 2009
84. 84. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 Sunday, March 15, 2009
85. 85. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 2 - 30 = -28 Sunday, March 15, 2009
86. 86. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 0 = x2(3x2 - 30x + 2x - 20) 2 - 30 = -28 Sunday, March 15, 2009
87. 87. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 0 = x2(3x2 - 30x + 2x - 20) 2 - 30 = -28 0 = x2[(3x2 - 30x) + (2x - 20)] Sunday, March 15, 2009
88. 88. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 0 = x2(3x2 - 30x + 2x - 20) 2 - 30 = -28 0 = x2[(3x2 - 30x) + (2x - 20)] 0 = x2[3x(x - 10) + 2(x - 10)] Sunday, March 15, 2009
89. 89. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 0 = x2(3x2 - 30x + 2x - 20) 2 - 30 = -28 0 = x2[(3x2 - 30x) + (2x - 20)] 0 = x2[3x(x - 10) + 2(x - 10)] 0 = x2(x - 10)(3x + 2) Sunday, March 15, 2009
90. 90. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 0 = x2(3x2 - 30x + 2x - 20) 2 - 30 = -28 0 = x2[(3x2 - 30x) + (2x - 20)] 0 = x2[3x(x - 10) + 2(x - 10)] 0 = x2(x - 10)(3x + 2) x=? Sunday, March 15, 2009
91. 91. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) 2(-30) = -60 0 = x2(3x2 - 30x + 2x - 20) 2 - 30 = -28 0 = x2[(3x2 - 30x) + (2x - 20)] 0 = x2[3x(x - 10) + 2(x - 10)] 0 = x2(x - 10)(3x + 2) x=? x = 0, 10, -2/3 Sunday, March 15, 2009
92. 92. Homework Sunday, March 15, 2009
93. 93. Homework p. 703 #2 - 27 Sunday, March 15, 2009
94. 94. Sunday, March 15, 2009