Special Products and Factoring Strategies

Review of Three Special Products

Recall the three special products:

  1. Difference of Squares

            x2 - y2  =  (x - y) (x + y)

  2. Square of Sum

            x2 + 2xy + y2  =  (x + y)2  

  3. 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.

  1. 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

     

  2. Now combine like terms to get  

            x3 - y3

Next, we state the sum of cubes formula.

  1.         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

  1.  36x2 - 4y2  =  (6x - 2y) (6x + 2y)       Notice that there only two terms.

  2. 3x3 - 12x2 + 12x  =  3x (x2 - 4x + 4)     Remember to pull the GCF out first.

    =  3x(x -2)2                  

  3. x6 - 64  =  (x3 - 8) (x3 + 8) 

    =  (x - 2) (x2 + 2x + 4) (x + 2) (x2 - 2x + 4)

 

Exercises:    

Factor the following

  1. 45a3b - 20ab3          5ab(3a - 2b)(3a + 2b)

  2. 64x6 - 16x3 + 1        (8x^3 - 1)^2

  3. x2 + 2xy + y2 - 81    (x + y - 9)(x + y + 9)

  4. x12 - y12     (Challenge Problem)  
     (x + y)(x - y)(x^2 + y^2)(x^2 +xy + y^2)(x^2 - xy + y^2)(x^4 - x^2y^2 + y^4)

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

  1. x3 - x                                    x(x - 1)(x + 1)

  2. x2 - 7x - 30                           (x - 10)(x + 3)

  3. 96a2 b - 48ab - 72a + 36       12(2a - 1)(4ab - 3)

  4. 4x2 - 36xy + 81y2                 (2x - 9y)^2

  5. 5a4b3 + 1080a                      5a(ab + 6)(a^2 b^2 - 6ab + 36)

  6. 2x2  + 5x - 12                       (x + 4)(2x - 3)

  7. 5x3 + 40                               5(x + 2)(x^2 - 2x + 4)

  8. x3 + 3x2 - 4x - 12                  (x - 2)(x + 2)(x + 3)

 

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