Section 11-5
                         The Factor Theorem




Sunday, March 15, 2009
In-Class Activity
      1. What were the x-intercepts for number 1? What was
      the factored form of the polynomial?


...
In-Class Activity
      1. What were the x-intercepts for number 1? What was
      the factored form of the polynomial?
  ...
In-Class Activity
      1. What were the x-intercepts for number 1? What was
      the factored form of the polynomial?
  ...
In-Class Activity
      1. What were the x-intercepts for number 1? What was
      the factored form of the polynomial?
  ...
In-Class Activity
      1. What were the x-intercepts for number 1? What was
      the factored form of the polynomial?
  ...
In-Class Activity
      1. What were the x-intercepts for number 1? What was
      the factored form of the polynomial?
  ...
In-Class Activity
      1. What were the x-intercepts for number 1? What was
      the factored form of the polynomial?
  ...
Zero-Product Theorem




Sunday, March 15, 2009
Zero-Product Theorem

                         For all a and b, ab = 0 IFF a = 0 or b = 0




Sunday, March 15, 2009
Zero-Product Theorem

                         For all a and b, ab = 0 IFF a = 0 or b = 0



            This means that i...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x
                          ...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
      a. Write a polynomial to represent the volume of the box.
          x
        x                         Le...
Example 1
                b. For what values of x is the volume exactly 0 in3?




Sunday, March 15, 2009
Example 1
                b. For what values of x is the volume exactly 0 in3?




Sunday, March 15, 2009
Example 1
                b. For what values of x is the volume exactly 0 in3?




Sunday, March 15, 2009
Example 1
                b. For what values of x is the volume exactly 0 in3?




Sunday, March 15, 2009
Example 1
                b. For what values of x is the volume exactly 0 in3?




Sunday, March 15, 2009
Sunday, March 15, 2009
Sunday, March 15, 2009
Sunday, March 15, 2009
Sunday, March 15, 2009
Sunday, March 15, 2009
x = 0, 10, 15



Sunday, March 15, 2009
Question:


      If there are two numbers that are being multiplied to get
      a product of 0, what can we say about at...
Factor Theorem

                    x - r is a factor of a polynomial P(x) IFF P(r) = 0




Sunday, March 15, 2009
Factor Theorem

                    x - r is a factor of a polynomial P(x) IFF P(r) = 0

      This means that if we have ...
Factor Theorem

                    x - r is a factor of a polynomial P(x) IFF P(r) = 0

      This means that if we have ...
Example 2
                         Find the zeros of P(x) = 3x3 - 33x2 + 90x
                              Set it equal to...
Example 2
                         Find the zeros of P(x) = 3x3 - 33x2 + 90x
                              Set it equal to...
Example 2
                         Find the zeros of P(x) = 3x3 - 33x2 + 90x
                              Set it equal to...
Example 2
                         Find the zeros of P(x) = 3x3 - 33x2 + 90x
                              Set it equal to...
Example 2
                         Find the zeros of P(x) = 3x3 - 33x2 + 90x
                              Set it equal to...
Example 2
                         Find the zeros of P(x) = 3x3 - 33x2 + 90x
                              Set it equal to...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Example 2
                           Find the zeros of P(x) = 3x3 - 33x2 + 90x
                                Set it equa...
Can we apply this to Example 1?
                         V(x) = 4x3 - 100x2 + 600x




Sunday, March 15, 2009
Can we apply this to Example 1?
                         V(x) = 4x3 - 100x2 + 600x

                          0 = 4x3 - 10...
Can we apply this to Example 1?
                         V(x) = 4x3 - 100x2 + 600x

                          0 = 4x3 - 10...
Can we apply this to Example 1?
                         V(x) = 4x3 - 100x2 + 600x

                          0 = 4x3 - 10...
Can we apply this to Example 1?
                                  V(x) = 4x3 - 100x2 + 600x

                             ...
Can we apply this to Example 1?
                                  V(x) = 4x3 - 100x2 + 600x

                             ...
Can we apply this to Example 1?
                                  V(x) = 4x3 - 100x2 + 600x

                             ...
Can we apply this to Example 1?
                                  V(x) = 4x3 - 100x2 + 600x

                             ...
Can we apply this to Example 1?
                                  V(x) = 4x3 - 100x2 + 600x

                             ...
Can we apply this to Example 1?
                                  V(x) = 4x3 - 100x2 + 600x

                             ...
Another question:


                         Why do we call these “zeros?”




Sunday, March 15, 2009
Another question:


                         Why do we call these “zeros?”


                          It’s where y is equ...
Yet another question:


                         What other names do we use for zeros?




Sunday, March 15, 2009
Yet another question:


                         What other names do we use for zeros?


                              Sol...
Example 3
                         Find P(x), which has zeros of -2, 0, and 2.




Sunday, March 15, 2009
Example 3
                         Find P(x), which has zeros of -2, 0, and 2.
                    Well, if we know the ze...
Example 3
                         Find P(x), which has zeros of -2, 0, and 2.
                    Well, if we know the ze...
Example 3
                         Find P(x), which has zeros of -2, 0, and 2.
                    Well, if we know the ze...
Example 3
                         Find P(x), which has zeros of -2, 0, and 2.
                    Well, if we know the ze...
Example 3
                         Find P(x), which has zeros of -2, 0, and 2.
                    Well, if we know the ze...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.




Sunday, March 15, 2009
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.

                                0 = x2(3x2 - 28x ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Example 4
                         Find the zeros of 3x4 - 28x3 - 20x2.
                                                  ...
Homework




Sunday, March 15, 2009
Homework



                         p. 703 #2 - 27




Sunday, March 15, 2009
Sunday, March 15, 2009
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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
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