Difference of Squares

There is a special formula for factoring a difference of squares that comes from applying a shortcut multiplication formula backwards. It turns out that whenever you multiply the sum and difference of the same things (hint: use the FOIL method), the Inners and Outers cancel out and you only get the First and Last terms:
                                (a + b)(a - b)   =   a2 - ab + ab - b2   =    a2 - b2

 

Written backwards, we get the formula for factoring:
 

                                  a2 - b2 = (a + b)(a - b)

 

So, if we see something that is a difference of squares, we can factor it as the sum and difference.

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

Example:              4x2 - 9 = (2x + 3)(2x - 3)
 

Sometimes we may need to factor out the GCF prior to factoring using this pattern.

Example:              6x2 - 24 = 6(x2 - 4) = 6(x - 2)(x + 2)

Example:              12x2 - 27 = 3(4x2 - 9) = 3(2x - 3)(2x + 3)
 

Sometimes we can use the formula more than once.

 

       x8 - 1 = (x4 + 1)(x4 - 1)   *** Note: Difference of Squares

 = (x4 + 1)(x2 + 1)(x2 - 1)   *** Note: Difference of Squares

 = (x4 + 1)(x2 + 1)(x - 1)(x + 1)

 

This brings up another point. Factoring is like prime factoring of numbers. Always factor completely, that is keep factoring the factors until nothing more will factor. The ones that can't factor come along for the ride. Sums of squares cannot be factored, so they come along for the ride.

 

Example:   x2  + 16 = x2  + 16   *** Note: This is an example of the "sum of two squares.  This is NOT factorable.

Additional explanation can be found on pages 469 - 470 of your textbook.  Do problems 12 - 16 on page 479.