Section 11-3
                          Factoring Special Cases




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




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

     1. Greatest Common Factor




Tuesday,...
Factoring:
                    Rewriting a polynomial as a product of factors

     1. Greatest Common Factor
     2. Bino...
Factoring:
                    Rewriting a polynomial as a product of factors

     1. Greatest Common Factor
     2. Bino...
Factoring:
                    Rewriting a polynomial as a product of factors

     1. Greatest Common Factor
     2. Bino...
Factoring:
                    Rewriting a polynomial as a product of factors

     1. Greatest Common Factor
     2. Bino...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2




Tuesday, March 3, 2009
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4




Tuesday, March 3, 20...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x




Tuesday, March 3, 2...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(




Tuesday, March 3, ...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3




Tuesday, March 3,...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x




Tuesday, March 3...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x -




Tuesday, March...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1




Tuesday, Mar...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)




Tuesday, Ma...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)       5




Tue...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)       5x




Tu...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)       5xy




T...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)       5xy(




...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)       5xy(3x2 +...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)       5xy(3x2 +...
Example 1: Factor.
                 a. 12x2 - 4x     b. 15x3y + 5x2y2 - 35xy2
                  4x(3x - 1)       5xy(3x2 +...
Binomial Square Factoring




Tuesday, March 3, 2009
Binomial Square Factoring

                         For all a and b:




Tuesday, March 3, 2009
Binomial Square Factoring

                            For all a and b:

                         a2 + 2ab + b2




Tuesda...
Binomial Square Factoring

                            For all a and b:

                         a2 + 2ab + b2 = (a + b)2...
Binomial Square Factoring

                            For all a and b:

                         a2 + 2ab + b2 = (a + b)2...
Binomial Square Factoring

                            For all a and b:

                         a2 + 2ab + b2 = (a + b)2...
Tuesday, March 3, 2009
(x +   4) 2




Tuesday, March 3, 2009
(x +   4) 2



                         (x + 4)(x + 4)




Tuesday, March 3, 2009
(x +   4) 2



                          (x + 4)(x + 4)

                          2   + 4x + 4x + 16
                    ...
(x +   4) 2



                          (x + 4)(x + 4)

                          2   + 4x + 4x + 16
                    ...
NOTICE
                           (x +   4) 2

                         First term:




Tuesday, March 3, 2009
NOTICE
                           (x +   4) 2

                         First term: x2




Tuesday, March 3, 2009
NOTICE
                                 (x +   4) 2

                               First term: x2
              What do y...
NOTICE
                                 (x +   4) 2

                               First term: x2
              What do y...
NOTICE
                                 (x +   4) 2

                               First term: x2
              What do y...
NOTICE
                                 (x +   4) 2

                               First term: x2
              What do y...
NOTICE
                                 (x +   4) 2

                               First term: x2
              What do y...
NOTICE
                                 (x +   4) 2

                               First term: x2
              What do y...
NOTICE
                                 (x +   4) 2

                               First term: x2
              What do y...
A pattern emerges...
      A perfect square trinomial will have the following things
                              occur:
...
A pattern emerges...
      A perfect square trinomial will have the following things
                              occur:
...
A pattern emerges...
      A perfect square trinomial will have the following things
                              occur:
...
A pattern emerges...
      A perfect square trinomial will have the following things
                              occur:
...
Example 2: Factor.
                              a. 9x2 + 12x + 4




Tuesday, March 3, 2009
Example 2: Factor.
                              a. 9x2 + 12x + 4



    Check to see if the first and last terms are perfe...
Example 2: Factor.
                              a. 9x2 + 12x + 4



    Check to see if the first and last terms are perfe...
Example 2: Factor.
                              a. 9x2 + 12x + 4

                           3x · 3x
    Check to see if ...
Example 2: Factor.
                              a. 9x2 + 12x + 4

                           3x · 3x
    Check to see if ...
Example 2: Factor.
                              a. 9x2 + 12x + 4

                           3x · 3x
    Check to see if ...
Example 2: Factor.
                              a. 9x2 + 12x + 4

                           3x · 3x      2·2
    Check t...
Example 2: Factor.
                              a. 9x2 + 12x + 4

                           3x · 3x      2·2
    Check t...
Example 2: Factor.
                              a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to ...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                               a. 9x2 + 12x + 4

                          3x · 3x      2·2
   Check to...
Example 2: Factor.
                         b. x2 - 6x + 9        c. y2 - 20y + 100




                                  ...
Example 2: Factor.
                         b. x2 - 6x + 9        c. y2 - 20y + 100
                           (x



     ...
Example 2: Factor.
                         b. x2 - 6x + 9        c. y2 - 20y + 100
                           (x 3)



  ...
Example 2: Factor.
                         b. x2 - 6x + 9        c. y2 - 20y + 100
                           (x - 3)



...
Example 2: Factor.
                         b. x2 - 6x + 9         c. y2 - 20y + 100
                           (x - 3) 2
...
Example 2: Factor.
                         b. x2 - 6x + 9         c. y2 - 20y + 100
                           (x - 3) 2 ...
Example 2: Factor.
                         b. x2 - 6x + 9         c. y2 - 20y + 100
                           (x - 3) 2 ...
Example 2: Factor.
                         b. x2 - 6x + 9         c. y2 - 20y + 100
                           (x - 3) 2 ...
Example 2: Factor.
                         b. x2 - 6x + 9         c. y2 - 20y + 100
                           (x - 3) 2 ...
Example 2: Factor.
                         b. x2 - 6x + 9          c. y2 - 20y + 100
                           (x - 3) 2...
Example 2: Factor.
                         b. x2 - 6x + 9          c. y2 - 20y + 100
                           (x - 3) 2...
Difference of Squares
                               Factoring




Tuesday, March 3, 2009
Difference of Squares
                               Factoring
                               For all a and b,




Tuesday...
Difference of Squares
                               Factoring
                               For all a and b,

          ...
Difference of Squares
                               Factoring
                               For all a and b,

          ...
Difference of two squares
                         This only works for the following conditions:




Tuesday, March 3, 2009
Difference of two squares
                         This only works for the following conditions:


                       ...
Difference of two squares
                         This only works for the following conditions:


                       ...
Difference of two squares
                         This only works for the following conditions:


                       ...
(t - 5)(t + 5)




Tuesday, March 3, 2009
(t - 5)(t + 5)




                         = t2




Tuesday, March 3, 2009
(t - 5)(t + 5)




                         = t2+ 5t




Tuesday, March 3, 2009
(t - 5)(t + 5)




                         = t2+ 5t - 5t




Tuesday, March 3, 2009
(t - 5)(t + 5)




                         = t2+ 5t - 5t - 25




Tuesday, March 3, 2009
(t - 5)(t + 5)




                         = t2+ 5t - 5t - 25




                             = t2 - 25


Tuesday, March...
Example 3: Factor.
                               a. 64x2 - 81




Tuesday, March 3, 2009
Example 3: Factor.
                               a. 64x2 - 81



    Check to see if the first and last terms are perfect ...
Example 3: Factor.
                               a. 64x2 - 81



    Check to see if the first and last terms are perfect ...
Example 3: Factor.
                               a. 64x2 - 81

                            8x · 8x
    Check to see if th...
Example 3: Factor.
                               a. 64x2 - 81

                            8x · 8x
    Check to see if th...
Example 3: Factor.
                               a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to see...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                                   a. 64x2 - 81

                            8x · 8x 9 · 9
    Check to...
Example 3: Factor.
                          b. r2 - 121        c. y2 + 100



                                           ...
Example 3: Factor.
                           b. r2 - 121       c. y2 + 100
                         (      )(     )


   ...
Example 3: Factor.
                           b. r2 - 121       c. y2 + 100
                         (r     )( r   )


   ...
Example 3: Factor.
                           b. r2 - 121       c. y2 + 100
                         (r 11)( r 11)


     ...
Example 3: Factor.
                           b. r2 - 121       c. y2 + 100
                         (r +11)( r - 11)


  ...
Example 3: Factor.
                           b. r2 - 121          c. y2 + 100
                         (r +11)( r - 11)  ...
Example 3: Factor.
                           b. r2 - 121          c. y2 + 100
                         (r +11)( r - 11)  ...
Example 3: Factor.
                           b. r2 - 121             c. y2 + 100
                         (r +11)( r - 11...
Example 3: Factor.
                           b. r2 - 121             c. y2 + 100
                         (r +11)( r - 11...
Example 3: Factor.
                           b. r2 - 121         c. y2 + 100
                         (r +11)( r - 11)   ...
Example 3: Factor.
                           b. r2 - 121         c. y2 + 100
                         (r +11)( r - 11)   ...
Example 3: Factor.
                           b. r2 - 121         c. y2 + 100
                         (r +11)( r - 11)   ...
Example 3: Factor.
                           b. r2 - 121         c. y2 + 100
                         (r +11)( r - 11)   ...
Example 3: Factor.
                           b. r2 - 121         c. y2 + 100
                         (r +11)( r - 11)   ...
Example 3: Factor.
                           b. r2 - 121         c. y2 + 100
                         (r +11)( r - 11)   ...
Example 3: Factor.
                           b. r2 - 121         c. y2 + 100
                         (r +11)( r - 11)   ...
Homework




Tuesday, March 3, 2009
Homework


                         p. 690 #1-12, 21, 22, 25




      “You must be the change you want to see in the worl...
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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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