Factoring the difference of two perfect squares
The formula for factoring the difference of two perfect squares
Factoring the difference of two squares is a special case of factoring a polynomial, where you’ll be factoring a binomial which is a difference of two terms that are both perfect squares.
For example, ???9x^2-16??? is the difference of two squares, because ???9x^2??? is the perfect square ???(3x)^2???, ???16??? is the perfect square ???4^2???, and ???16??? is being subtracted from ???9x^2???.
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To factor a difference of two squares, you simply take the expressions that are squared (in the example the expressions ???3x??? and ???4???) and put both of them (in the given order) into the terms in both factors of the given binomial. In one factor, you’ll add the second expression to the first one; in the other factor, you’ll subtract the second expression from the first one. Therefore, ???9x^2-16??? is factored as ???(3x+4)(3x-4)???.
Examples of difference of squares factoring
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Factoring a perfect square binomial
Example
Factor the binomial.
???x^2-25???
Since ???x^2??? and ???25??? are both perfect squares (the squares of ???x??? and ???5???, respectively), ???x^2-25??? is factored as
???(x+5)(x-5)???
Let’s try another example of factoring the difference of two squares.
In one factor, you’ll add the second expression to the first one; in the other factor, you’ll subtract the second expression from the first one.
Example
Factor the binomial.
???64x^4y^2-9z^6???
Notice that ???64x^4y^2=(8x^2y)^2???, and that ???9z^6=(3z^3)^2???. Therefore, ???64x^4y^2-9z^6??? is factored as
???(8x^2y+3z^3)(8x^2y-3z^3)???
