SOLUTION: How do you determine if a polynomial is the difference of two squares?

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Question 135231: How do you determine if a polynomial is the difference
of two squares?
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Difference means one thing subtracted from the other. The difference of two squares is then one squared thing subtracted from another squared thing.

For example: a%5E2 is a squared thing, and b%5E2 is a squared thing, so a%5E2-b%5E2 is the difference of two squares.

x%5E2-64 -- 64 is 8%5E2, so x%5E2-64=%28x%2B8%29%28x-8%29

r%5E2-2 -- 2 is %28sqrt%282%29%29%5E2, so r%5E2-2=%28r%2Bsqrt%282%29%29%28r-sqrt%282%29%29

The fact is, any binomial (two term polynomial) with opposite signs on the terms (+ on one, - on the other) is the difference of two squares if you are willing to put up with radicals in your factorization. And if you don't mind factoring over the complex numbers, you don't even have to worry about the signs restriction.

%28x-3%29=%28sqrt%28x%29%2Bsqrt%283%29%29%28sqrt%28x%29-sqrt%283%29%29

%28x%2B3%29=%28sqrt%28x%29%2Bi%2Asqrt%283%29%29%28sqrt%28x%29-i%2Asqrt%283%29%29 where i%5E2=-1

Typically, however, we consider the difference of two squares to be the difference of two PERFECT squares. The way to quickly recognize the difference of two squares pattern is to memorize your table of perfect squares -- or carry a table of squares around with you.