Special Products and Factoring Strategies Review of Three Special Products Recall the three special products: Difference of Squares         x2 - y2  =  (x - y) (x + y) Square of Sum         x2 + 2xy + y2  =  (x + y)2   Square of Difference         x2 - 2xy + y2  =  (x - y)2 Special Products Involving Cubes Just as there is a difference of squares formula, there is also a difference of cubes formula. x3 - y3 = (x - y) (x2 + xy + y2) Proof:   We use the distributive law on the right hand side         x (x2 + xy + y2) - y (x2 + xy + y2)          =   x3 + x2y + xy2 - x2y - xy2 - y3   Now combine like terms to get           x3 - y3 Next, we state the sum of cubes formula.         x3 + y3  =  (x + y)(x2 - xy + y2) Exercise Prove the sum of cubes equation (Equation 5)    Using the Special Product Formulas for Factoring   Examples:  Factor the following  36x2 - 4y2  =  (6x - 2y) (6x + 2y)       Notice that there only two terms. 3x3 - 12x2 + 12x  =  3x (x2 - 4x + 4)     Remember to pull the GCF out first. =  3x(x -2)2               x6 - 64  =  (x3 - 8) (x3 + 8)  =  (x - 2) (x2 + 2x + 4) (x + 2) (x2 - 2x + 4)   Exercises:     Factor the following 45a3b - 20ab3          64x6 - 16x3 + 1        x2 + 2xy + y2 - 81    x12 - y12     (Challenge Problem)     Factoring Strategies Always pull out the GCF first Look for special products.  If there are only two terms then look for sum of cubes or difference of squares or cubes.  If there are three terms, look for squares of a difference or a sum. If there are three terms and the first coefficient is 1 then use simple trinomial factoring. If there are three terms and the first coefficient is not 1 then use the AC method. If there are four terms then try factoring by grouping.   Exercises x3 - x                                    x2 - 7x - 30                           96a2 b - 48ab - 72a + 36       4x2 - 36xy + 81y2                 5a4b3 + 1080a                      2x2  + 5x - 12                       5x3 + 40                               x3 + 3x2 - 4x - 12