Difference of squares in disguise
SAT glossary · Course Foundations
Definition
a^2-b^2=(a+b)(a-b) is the most tested factoring pattern on the examination, and it is tested disguised: 4x^2-9=(2x+3)(2x-3); x^4-16=(x^2+4)(x+2)(x-2) — students routinely stop short of the second factoring; and x^2-5=(x+sqrt5)(x-sqrt5), since nothing requires b to be an integer. There is no sum-of-squares factoring over the reals, and the examination offers one as a distractor.
Difference of squares in disguise in plain English
The pattern a^2-b^2=(a+b)(a-b) — called 'difference of squares' — is probably the single most useful factoring shortcut on the whole test, and it loves to hide in disguise. Sometimes it looks obvious, like 4x^2-9, which factors into (2x+3)(2x-3). Sometimes it needs a second round — x^4-16 factors first into (x^2+4)(x^2-4), but that second piece is ITSELF a difference of squares and needs to be broken down again into (x+2)(x-2), so the fully factored form is (x^2+4)(x+2)(x-2). And sometimes b isn't even a whole number: x^2-5 still factors, into (x+sqrt5)(x-sqrt5), since sqrt5 squared just gives you 5 back. One important warning: there is NO equivalent shortcut for a SUM of squares like a^2+b^2 — that expression simply doesn't factor over real numbers, no matter how tempting it looks, so don't fall for an answer choice that pretends it does.
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