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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

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