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Difference of Squares

If a polynomial can be written as a 2 − b 2 , then it can be factored as a difference of squares :

a 2 − b 2 = ( a + b ) ( a − b )

This is because, when you use FOIL to expand the right side, the a b terms cancel out:

( a + b ) ( a − b ) = a 2 − a b + a b − b 2

Example 1:

Factor, if possible.

x 2 − 64

This is a difference of squares with a = x and b = 8 .

x 2 − 64 = ( x − 8 ) ( x + 8 )

Example 2:

Factor, if possible.

9 p 2 − 49 q 2

This is a difference of squares with a = 3 p and b = 7 q .

9 p 2 − 49 q 2 = ( 3 p − 7 q ) ( 3 p + 7 q )

You should verify these equalities using FOIL .