Difference of Squares

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In the Investigation, you expanded two expressions that form a special group of polynomials called a difference of squares. In both cases, when expanded using the distributive property, the middle two terms from FOIL were eliminated. The result was the product of the first terms subtract the product of the last terms. The key features of the difference of squares of these polynomials is that there are only two terms in the polynomial, both terms are perfect squares, and the second term is subtracted from the first term.


It is important to recognize these special polynomials because it will save a lot of time! While they can be factored using the methods from Sections B and C, they can be factored more quickly if they are recognized.

Once recognized as a difference of squares, factoring is done by first taking the square root of both terms. Then, in one factor you add the square root of each term; and in the second factor you subtract the square root of each term. Note that order does not matter; you can subtract in the first factor and add in the second factor.

\(a^2 x^2 - b^2 y^2 = \left( {ax + by} \right)\left( {ax - by} \right)\)
\(a^2 \left( {f\left( x \right)} \right)^2 - b^2 \left( {g\left( y \right)} \right)^2 = \left[ {a\left( {f\left( x \right)} \right) + b\left( {g\left( y \right)} \right)} \right]\left[ {a\left( {f\left( x \right)} \right) - b\left( {g\left( y \right)} \right)} \right]\)