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# AA Section 11-3 Day 1

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Factoring Special Cases

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### AA Section 11-3 Day 1

1. 1. Section 11-3 Factoring Special Cases Tuesday, March 3, 2009
2. 2. Factoring: Rewriting a polynomial as a product of factors Tuesday, March 3, 2009
3. 3. Factoring: Rewriting a polynomial as a product of factors 1. Greatest Common Factor Tuesday, March 3, 2009
4. 4. Factoring: Rewriting a polynomial as a product of factors 1. Greatest Common Factor 2. Binomial Square Factoring Tuesday, March 3, 2009
5. 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. 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. 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. 8. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 Tuesday, March 3, 2009
9. 9. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4 Tuesday, March 3, 2009
10. 10. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x Tuesday, March 3, 2009
11. 11. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x( Tuesday, March 3, 2009
12. 12. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3 Tuesday, March 3, 2009
13. 13. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x Tuesday, March 3, 2009
14. 14. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - Tuesday, March 3, 2009
15. 15. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1 Tuesday, March 3, 2009
16. 16. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) Tuesday, March 3, 2009
17. 17. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5 Tuesday, March 3, 2009
18. 18. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5x Tuesday, March 3, 2009
19. 19. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5xy Tuesday, March 3, 2009
20. 20. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5xy( Tuesday, March 3, 2009
21. 21. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5xy(3x2 + xy - 7y) Tuesday, March 3, 2009
22. 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. 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. 24. Binomial Square Factoring Tuesday, March 3, 2009
25. 25. Binomial Square Factoring For all a and b: Tuesday, March 3, 2009
26. 26. Binomial Square Factoring For all a and b: a2 + 2ab + b2 Tuesday, March 3, 2009
27. 27. Binomial Square Factoring For all a and b: a2 + 2ab + b2 = (a + b)2 Tuesday, March 3, 2009
28. 28. Binomial Square Factoring For all a and b: a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 Tuesday, March 3, 2009
29. 29. Binomial Square Factoring For all a and b: a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2 Tuesday, March 3, 2009
30. 30. Tuesday, March 3, 2009
31. 31. (x + 4) 2 Tuesday, March 3, 2009
32. 32. (x + 4) 2 (x + 4)(x + 4) Tuesday, March 3, 2009
33. 33. (x + 4) 2 (x + 4)(x + 4) 2 + 4x + 4x + 16 x Tuesday, March 3, 2009
34. 34. (x + 4) 2 (x + 4)(x + 4) 2 + 4x + 4x + 16 x 2 + 8x + 16 x Tuesday, March 3, 2009
35. 35. NOTICE (x + 4) 2 First term: Tuesday, March 3, 2009
36. 36. NOTICE (x + 4) 2 First term: x2 Tuesday, March 3, 2009
37. 37. NOTICE (x + 4) 2 First term: x2 What do you notice about it compared to what we began with? Tuesday, March 3, 2009
38. 38. NOTICE (x + 4) 2 First term: x2 What do you notice about it compared to what we began with? Middle term: Tuesday, March 3, 2009
39. 39. NOTICE (x + 4) 2 First term: x2 What do you notice about it compared to what we began with? Middle term: 8x Tuesday, March 3, 2009
40. 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? Tuesday, March 3, 2009
41. 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: Tuesday, March 3, 2009
42. 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. 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? Tuesday, March 3, 2009
44. 44. A pattern emerges... A perfect square trinomial will have the following things occur: Tuesday, March 3, 2009
45. 45. A pattern emerges... A perfect square trinomial will have the following things occur: 1. The ﬁrst term will be a perfect square. Tuesday, March 3, 2009
46. 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. 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. 48. Example 2: Factor. a. 9x2 + 12x + 4 Tuesday, March 3, 2009
49. 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. 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. 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. 52. 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
53. 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. 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 Tuesday, March 3, 2009
55. 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 Tuesday, March 3, 2009
56. 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. 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. 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. 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. 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: Tuesday, March 3, 2009
61. 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. 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. 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. 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 Tuesday, March 3, 2009
65. 65. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 d. x2 + 7x + 14 Tuesday, March 3, 2009
66. 66. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x d. x2 + 7x + 14 Tuesday, March 3, 2009
67. 67. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x 3) d. x2 + 7x + 14 Tuesday, March 3, 2009
68. 68. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) d. x2 + 7x + 14 Tuesday, March 3, 2009
69. 69. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) 2 d. x2 + 7x + 14 Tuesday, March 3, 2009
70. 70. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) 2 (y d. x2 + 7x + 14 Tuesday, March 3, 2009
71. 71. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) 2 (y - d. x2 + 7x + 14 Tuesday, March 3, 2009
72. 72. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) 2 (y - 10) d. x2 + 7x + 14 Tuesday, March 3, 2009
73. 73. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) 2 (y - 10)2 d. x2 + 7x + 14 Tuesday, March 3, 2009
74. 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. 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. 76. Difference of Squares Factoring Tuesday, March 3, 2009
77. 77. Difference of Squares Factoring For all a and b, Tuesday, March 3, 2009
78. 78. Difference of Squares Factoring For all a and b, a2 - b2 = Tuesday, March 3, 2009
79. 79. Difference of Squares Factoring For all a and b, a2 - b2 = (a + b)(a - b) Tuesday, March 3, 2009
80. 80. Difference of two squares This only works for the following conditions: Tuesday, March 3, 2009
81. 81. Difference of two squares This only works for the following conditions: 1. You must have a binomial. Tuesday, March 3, 2009
82. 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. 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! Tuesday, March 3, 2009
84. 84. (t - 5)(t + 5) Tuesday, March 3, 2009
85. 85. (t - 5)(t + 5) = t2 Tuesday, March 3, 2009
86. 86. (t - 5)(t + 5) = t2+ 5t Tuesday, March 3, 2009
87. 87. (t - 5)(t + 5) = t2+ 5t - 5t Tuesday, March 3, 2009
88. 88. (t - 5)(t + 5) = t2+ 5t - 5t - 25 Tuesday, March 3, 2009
89. 89. (t - 5)(t + 5) = t2+ 5t - 5t - 25 = t2 - 25 Tuesday, March 3, 2009
90. 90. Example 3: Factor. a. 64x2 - 81 Tuesday, March 3, 2009
91. 91. Example 3: Factor. a. 64x2 - 81 Check to see if the ﬁrst and last terms are perfect squares. Tuesday, March 3, 2009
92. 92. Example 3: Factor. a. 64x2 - 81 Check to see if the ﬁrst and last terms are perfect squares. Tuesday, March 3, 2009
93. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 104. Example 3: Factor. b. r2 - 121 c. y2 + 100 e. x4 - 16 d. 25x4y6 - 36z8 Tuesday, March 3, 2009
105. 105. Example 3: Factor. b. r2 - 121 c. y2 + 100 ( )( ) e. x4 - 16 d. 25x4y6 - 36z8 Tuesday, March 3, 2009
106. 106. Example 3: Factor. b. r2 - 121 c. y2 + 100 (r )( r ) e. x4 - 16 d. 25x4y6 - 36z8 Tuesday, March 3, 2009
107. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 120. Homework Tuesday, March 3, 2009
121. 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