I hope you remember the general rule, or shortcut we developed to handle special products like these. When the terms are exactly the same and only the signs in the parenthesis are different, the result is called The Difference of Two Squares. This time, we’ll go in the opposite direction. We’ll start with the difference of two squares and work out what the factors must be to get that result. By the end of the lesson you should be able to look at the difference of two squares and write the factors at once, without any working in between.
Let’s do an example. Factorize X squared minus nine. First, always check to see if there are any common factors. In this one, there are no common factors. The next thing to do is to see if the two terms can be written as the difference of two squares. X squared is a square, obviously, but can nine be written as a square? That’s right, nine is three squared. So we have X squared minus three squared. That means, our terms for both factors are X and three. Now we just have to put a minus sign in one set of parenthesis and a plus sign in the other and we’re done. X minus three and X plus three are the factors of X squared minus nine. Remember, the signs can be the other way around, X plus three, X minus three.
So, if you’re asked to factorize an expression that has two terms, first check to see if there are any common factors. If not, see if it can be written as a difference of two squares.