The Factor Theorem

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