# How to Factor a Perfect Square

FOIL stands for multiply the *first, outside, inside,* and *last* terms together. When you FOIL a binomial times itself, the product is called a *perfect square.* For example, (*a* + *b*)^{2} gives you the perfect-square trinomial *a*^{2} + 2*ab* + *b*^{2}. Because a perfect-square trinomial is still a trinomial, you follow the steps in the backward FOIL method of factoring. However,* *you must account for one extra step at the very end where you express the answer as a binomial squared.

For example, to factor the polynomial 4*x*^{2} – 12*x *+ 9, follow these steps:

Multiply the quadratic term and the constant term.

The product of the quadratic term 4

*x*^{2}and the constant 9 is 36*x*^{2}, so that made your job easy.Write down all the factors of the result that result in pairs in which each term in the pair has an

*x*.Following are the factors of 36

*x*^{2}in pairs:1

*x*and 36*x*–1

*x*and –36*x*2

*x*and 18*x*–2

*x*and –18*x*3

*x*and 12*x*–3

*x*and –12*x*4

*x*and 9*x*–4

*x*and –9*x*6

*x*and 6*x*–6

*x*and –6*x*

If you think ahead to the next step, you can skip writing out the positive factors, because they produce only

*x*terms with a positive coefficient.From this list, find the pair that adds to produce the coefficient of the linear term.

You want to get a sum of –12

*x*in this case. The only way to do that is to use –6*x*and –6*x.*Break up the linear term into two terms, using the terms from Step 3.

You now get 4

*x*^{2}– 6*x*– 6*x*+ 9.Group the four terms into two sets of two.

Remember to include the plus sign between the two groups, resulting in (4

*x*^{2}– 6*x*) + (–6*x*+ 9).Find the Greatest Common Factor (GCF) for each set and factor it out.

The GCF of the first two terms is 2

*x,*and the GCF of the next two terms is –3; when you factor them out, you get 2*x*(2*x*– 3) – 3(2*x*– 3).Find the GCF of the two new terms.

This time the GCF is (2

*x*– 3); when you factor it out, you get (2*x*– 3)(2*x*– 3). Aha! That’s a binomial times itself, which means you have one extra step.Express the resulting product as a binomial squared.

This step is easy: (2

*x*– 3)^{2}.