Special Factoring: Differences of Squares

For this quadratic factorization, I need factors of −4 that add up to zero, so I'll use −2 and +2:

(Review Factoring Quadratics, if the steps in this example didn't make sense to you.)

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Note that we had x² − 2², and ended up with (x − 2)(x + 2). Differences of squares (being something squared minus something else squared) always work this way:

For a² − b², I start by doing the parentheses:

Then I put the first squared thing in front:

...and I put the second squared thing in back:

...and then I alternate the signs in the middles:

Because the factoring always works out exactly the same way, we can turn it into a formula:

Memorize this formula! It will come in handy later, especially when you get to rational expressions (polynomial fractions). And you'll probably be expected to know this formula for your next test.

By the way, no, the order of the factors doesn't matter. Since multiplication is commutative (that is, since you can move the factors around without changing the value of the product), the difference of squares can also be stated as:

Don't get hung up on the order of the factors. Either way is fine.

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Here are examples of some typical homework problems:

• Factor x² − 16

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• Factor 4x² − 25

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• Factor 9x⁶ − y⁸

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• Factor x⁴ − 1

The answer to this last exercise depended on the fact that 1, to any power at all, is still just 1.

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Warning: Never forget that this formula is for the difference of squares (with variables); the polynomial sum of squares is always prime (that is, it can't be factored with whole numbers or fractions).