AA Section 11-5
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The Factor Theorem

The Factor Theorem

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AA Section 11-5 Presentation Transcript

  • 1. Section 11-5 The Factor Theorem Sunday, March 15, 2009
  • 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. 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. 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. 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. 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. 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. 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. Zero-Product Theorem Sunday, March 15, 2009
  • 10. Zero-Product Theorem For all a and b, ab = 0 IFF a = 0 or b = 0 Sunday, March 15, 2009
  • 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. Example 1 a. Write a polynomial to represent the volume of the box. x x 20 in. 30 in. Sunday, March 15, 2009
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
  • 28. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
  • 29. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
  • 30. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
  • 31. Example 1 b. For what values of x is the volume exactly 0 in3? Sunday, March 15, 2009
  • 32. Sunday, March 15, 2009
  • 33. Sunday, March 15, 2009
  • 34. Sunday, March 15, 2009
  • 35. Sunday, March 15, 2009
  • 36. Sunday, March 15, 2009
  • 37. x = 0, 10, 15 Sunday, March 15, 2009
  • 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. Factor Theorem x - r is a factor of a polynomial P(x) IFF P(r) = 0 Sunday, March 15, 2009
  • 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. 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. Example 2 Find the zeros of P(x) = 3x3 - 33x2 + 90x Set it equal to 0 and factor it! Sunday, March 15, 2009
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x Sunday, March 15, 2009
  • 62. Can we apply this to Example 1? V(x) = 4x3 - 100x2 + 600x 0 = 4x3 - 100x2 + 600x Sunday, March 15, 2009
  • 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. 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. 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. 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. 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. 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. 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. 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. Another question: Why do we call these “zeros?” Sunday, March 15, 2009
  • 72. Another question: Why do we call these “zeros?” It’s where y is equal to zero. Sunday, March 15, 2009
  • 73. Yet another question: What other names do we use for zeros? Sunday, March 15, 2009
  • 74. Yet another question: What other names do we use for zeros? Solutions, x-intercepts, roots Sunday, March 15, 2009
  • 75. Example 3 Find P(x), which has zeros of -2, 0, and 2. Sunday, March 15, 2009
  • 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. 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. 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. 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. 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. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. Sunday, March 15, 2009
  • 82. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 0 = x2(3x2 - 28x - 20) Sunday, March 15, 2009
  • 83. Example 4 Find the zeros of 3x4 - 28x3 - 20x2. 3(-20) = -60 0= x2(3x2 - 28x - 20) Sunday, March 15, 2009
  • 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. 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. 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. 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. 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. 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. 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. 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. Homework Sunday, March 15, 2009
  • 93. Homework p. 703 #2 - 27 Sunday, March 15, 2009
  • 94. Sunday, March 15, 2009