Factoring Special Cases

1. Section 11-3
Factoring Special Cases
Tuesday, March 3, 2009

2. Factoring:
Rewriting a polynomial as a product of factors
Tuesday, March 3, 2009

3. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
Tuesday, March 3, 2009

4. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
Tuesday, March 3, 2009

5. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
3. Difference of Squares Factoring
Tuesday, March 3, 2009

6. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
3. Difference of Squares Factoring
4. Other Methods of Factoring
Tuesday, March 3, 2009

7. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
3. Difference of Squares Factoring
4. Other Methods of Factoring
There’s trial-and-error, too, but that just takes too long.
Tuesday, March 3, 2009

8. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
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9. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4
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10. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x
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11. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(
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12. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3
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13. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x
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14. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x -
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15. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1
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16. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1)
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17. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5
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18. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5x
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19. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy
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20. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(
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21. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(3x2 + xy - 7y)
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22. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(3x2 + xy - 7y)
All we did here was go through the numbers first, then the
variables in alphabetical order, finding factors that the terms
have in common.
Tuesday, March 3, 2009

23. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(3x2 + xy - 7y)
All we did here was go through the numbers first, then the
variables in alphabetical order, finding factors that the terms
have in common.
To check your answer, re-distribute the GCF and see if you
get what you started with.
Tuesday, March 3, 2009

24. Binomial Square Factoring
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25. Binomial Square Factoring
For all a and b:
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26. Binomial Square Factoring
For all a and b:
a2 + 2ab + b2
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27. Binomial Square Factoring
For all a and b:
a2 + 2ab + b2 = (a + b)2
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28. Binomial Square Factoring
For all a and b:
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2
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29. Binomial Square Factoring
For all a and b:
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
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30. Tuesday, March 3, 2009

31. (x + 4) 2
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32. (x + 4) 2
(x + 4)(x + 4)
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33. (x + 4) 2
(x + 4)(x + 4)
2 + 4x + 4x + 16
x
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34. (x + 4) 2
(x + 4)(x + 4)
2 + 4x + 4x + 16
x
2 + 8x + 16
x
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35. NOTICE
(x + 4) 2
First term:
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36. NOTICE
(x + 4) 2
First term: x2
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37. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
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38. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term:
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39. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
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40. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
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41. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
Last term:
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42. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
Last term: 16
Tuesday, March 3, 2009

43. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
Last term: 16
What’s happening?
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44. A pattern emerges...
A perfect square trinomial will have the following things
occur:
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45. A pattern emerges...
A perfect square trinomial will have the following things
occur:
1. The first term will be a perfect square.
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46. A pattern emerges...
A perfect square trinomial will have the following things
occur:
1. The first term will be a perfect square.
2.The last term will be a perfect square.
Tuesday, March 3, 2009

47. A pattern emerges...
A perfect square trinomial will have the following things
occur:
1. The first term will be a perfect square.
2.The last term will be a perfect square.
3.The middle term will be 2 times the product of the square
roots of the first and last terms.
Tuesday, March 3, 2009

48. Example 2: Factor.
a. 9x2 + 12x + 4
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49. Example 2: Factor.
a. 9x2 + 12x + 4
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

50. Example 2: Factor.
a. 9x2 + 12x + 4
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

51. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

52. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x
Check to see if the first and last terms are perfect squares.
3x
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53. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x
Check to see if the first and last terms are perfect squares.
3x
Tuesday, March 3, 2009

54. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x
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55. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
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56. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
Tuesday, March 3, 2009

57. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2)
Tuesday, March 3, 2009

58. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x)
Tuesday, March 3, 2009

59. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Tuesday, March 3, 2009

60. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
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61. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x
Tuesday, March 3, 2009

62. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x 2)
Tuesday, March 3, 2009

63. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x + 2)
Tuesday, March 3, 2009

64. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x + 2)2
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65. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
d. x2 + 7x + 14
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66. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x
d. x2 + 7x + 14
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67. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x 3)
d. x2 + 7x + 14
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68. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3)
d. x2 + 7x + 14
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69. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2
d. x2 + 7x + 14
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70. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y
d. x2 + 7x + 14
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71. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y -
d. x2 + 7x + 14
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72. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)
d. x2 + 7x + 14
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73. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)2
d. x2 + 7x + 14
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74. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)2
d. x2 + 7x + 14
(x
Tuesday, March 3, 2009

75. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)2
d. x2 + 7x + 14
(x
14 is not a perfect square!
Cannot factor with this method.
Tuesday, March 3, 2009

76. Difference of Squares
Factoring
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77. Difference of Squares
Factoring
For all a and b,
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78. Difference of Squares
Factoring
For all a and b,
a2 - b2 =
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79. Difference of Squares
Factoring
For all a and b,
a2 - b2 =
(a + b)(a - b)
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80. Difference of two squares
This only works for the following conditions:
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81. Difference of two squares
This only works for the following conditions:
1. You must have a binomial.
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82. Difference of two squares
This only works for the following conditions:
1. You must have a binomial.
2.Both terms must be perfect squares.
Tuesday, March 3, 2009

83. Difference of two squares
This only works for the following conditions:
1. You must have a binomial.
2.Both terms must be perfect squares.
3.There must be subtraction!
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84. (t - 5)(t + 5)
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85. (t - 5)(t + 5)
= t2
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86. (t - 5)(t + 5)
= t2+ 5t
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87. (t - 5)(t + 5)
= t2+ 5t - 5t
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88. (t - 5)(t + 5)
= t2+ 5t - 5t - 25
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89. (t - 5)(t + 5)
= t2+ 5t - 5t - 25
= t2 - 25
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90. Example 3: Factor.
a. 64x2 - 81
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91. Example 3: Factor.
a. 64x2 - 81
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

92. Example 3: Factor.
a. 64x2 - 81
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

93. Example 3: Factor.
a. 64x2 - 81
8x · 8x
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

94. Example 3: Factor.
a. 64x2 - 81
8x · 8x
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

95. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009

96. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Tuesday, March 3, 2009

97. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
Tuesday, March 3, 2009

98. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x
Tuesday, March 3, 2009

99. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x (8x
Tuesday, March 3, 2009

100. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x 9)(8x
Tuesday, March 3, 2009

101. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x 9)(8x 9)
Tuesday, March 3, 2009

102. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x + 9)(8x 9)
Tuesday, March 3, 2009

103. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x + 9)(8x - 9)
Tuesday, March 3, 2009

104. Example 3: Factor.
b. r2 - 121 c. y2 + 100
e. x4 - 16
d. 25x4y6 - 36z8
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105. Example 3: Factor.
b. r2 - 121 c. y2 + 100
( )( )
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009

106. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r )( r )
e. x4 - 16
d. 25x4y6 - 36z8
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107. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r 11)( r 11)
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009

108. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11)
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009

109. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009

110. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009

111. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6 - 36z8
( )( )
Tuesday, March 3, 2009

112. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6 -36z8
(5x2y3 )(5x2y3 )
Tuesday, March 3, 2009

113. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3 6z4)(5x2y3 6z4 )
Tuesday, March 3, 2009

114. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 )
Tuesday, March 3, 2009

115. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) ( )( )
Tuesday, March 3, 2009

116. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2 )( x2 )
Tuesday, March 3, 2009

117. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2 4)( x2 4 )
Tuesday, March 3, 2009

118. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2+ 4)( x2 - 4 )
Tuesday, March 3, 2009

119. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2+ 4)( x2 - 4 )
(x2 + 4)(x + 2)(x - 2)
Tuesday, March 3, 2009

120. Homework
Tuesday, March 3, 2009

121. Homework
p. 690 #1-12, 21, 22, 25
“You must be the change you want to see in the world” -
Mahatma Ghandi
Tuesday, March 3, 2009