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Factoring Special Cases

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- 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 Tuesday, March 3, 2009
- 9. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4 Tuesday, March 3, 2009
- 10. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x Tuesday, March 3, 2009
- 11. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x( Tuesday, March 3, 2009
- 12. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3 Tuesday, March 3, 2009
- 13. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x Tuesday, March 3, 2009
- 14. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - Tuesday, March 3, 2009
- 15. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1 Tuesday, March 3, 2009
- 16. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) Tuesday, March 3, 2009
- 17. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5 Tuesday, March 3, 2009
- 18. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5x Tuesday, March 3, 2009
- 19. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5xy Tuesday, March 3, 2009
- 20. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5xy( Tuesday, March 3, 2009
- 21. Example 1: Factor. a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2 4x(3x - 1) 5xy(3x2 + xy - 7y) Tuesday, March 3, 2009
- 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 Tuesday, March 3, 2009
- 25. Binomial Square Factoring For all a and b: Tuesday, March 3, 2009
- 26. Binomial Square Factoring For all a and b: a2 + 2ab + b2 Tuesday, March 3, 2009
- 27. Binomial Square Factoring For all a and b: a2 + 2ab + b2 = (a + b)2 Tuesday, March 3, 2009
- 28. Binomial Square Factoring For all a and b: a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 Tuesday, March 3, 2009
- 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. Tuesday, March 3, 2009
- 31. (x + 4) 2 Tuesday, March 3, 2009
- 32. (x + 4) 2 (x + 4)(x + 4) Tuesday, March 3, 2009
- 33. (x + 4) 2 (x + 4)(x + 4) 2 + 4x + 4x + 16 x Tuesday, March 3, 2009
- 34. (x + 4) 2 (x + 4)(x + 4) 2 + 4x + 4x + 16 x 2 + 8x + 16 x Tuesday, March 3, 2009
- 35. NOTICE (x + 4) 2 First term: Tuesday, March 3, 2009
- 36. NOTICE (x + 4) 2 First term: x2 Tuesday, March 3, 2009
- 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. 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. 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. 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. 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. 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? Tuesday, March 3, 2009
- 44. A pattern emerges... A perfect square trinomial will have the following things occur: Tuesday, March 3, 2009
- 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. 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 Tuesday, March 3, 2009
- 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 Tuesday, March 3, 2009
- 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 Tuesday, March 3, 2009
- 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. 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: Tuesday, March 3, 2009
- 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 Tuesday, March 3, 2009
- 65. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 d. x2 + 7x + 14 Tuesday, March 3, 2009
- 66. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x d. x2 + 7x + 14 Tuesday, March 3, 2009
- 67. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x 3) d. x2 + 7x + 14 Tuesday, March 3, 2009
- 68. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) d. x2 + 7x + 14 Tuesday, March 3, 2009
- 69. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) 2 d. x2 + 7x + 14 Tuesday, March 3, 2009
- 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. Example 2: Factor. b. x2 - 6x + 9 c. y2 - 20y + 100 (x - 3) 2 (y - d. x2 + 7x + 14 Tuesday, March 3, 2009
- 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. 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. 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 Tuesday, March 3, 2009
- 77. Difference of Squares Factoring For all a and b, Tuesday, March 3, 2009
- 78. Difference of Squares Factoring For all a and b, a2 - b2 = Tuesday, March 3, 2009
- 79. Difference of Squares Factoring For all a and b, a2 - b2 = (a + b)(a - b) Tuesday, March 3, 2009
- 80. Difference of two squares This only works for the following conditions: Tuesday, March 3, 2009
- 81. Difference of two squares This only works for the following conditions: 1. You must have a binomial. Tuesday, March 3, 2009
- 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! Tuesday, March 3, 2009
- 84. (t - 5)(t + 5) Tuesday, March 3, 2009
- 85. (t - 5)(t + 5) = t2 Tuesday, March 3, 2009
- 86. (t - 5)(t + 5) = t2+ 5t Tuesday, March 3, 2009
- 87. (t - 5)(t + 5) = t2+ 5t - 5t Tuesday, March 3, 2009
- 88. (t - 5)(t + 5) = t2+ 5t - 5t - 25 Tuesday, March 3, 2009
- 89. (t - 5)(t + 5) = t2+ 5t - 5t - 25 = t2 - 25 Tuesday, March 3, 2009
- 90. Example 3: Factor. a. 64x2 - 81 Tuesday, March 3, 2009
- 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 Tuesday, March 3, 2009
- 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 Tuesday, March 3, 2009
- 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

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