Math Tip of the Week – Multiplying and Factoring Polynomials

A couple of techniques that you’ll have to learn early in your high school math career is multiplying and factoring polynomials.  Polynomials are a collection of coefficients and variables with exponents added and subtracted together, but none of the variables appear in denominators.

You may have heard of the “foiling” process, which they actually don’t teach as much anymore.  Nevertheless, I still feel it’s very important to learn, since it’s then easier to “unfoil” (factor):


FOIL stands for First Outer Inner Last.  Just remember that we multiply the First Outer Inner and then Last terms and put plus signs between them (unless the product is negative).  Here are some examples.   Note that FOILing only works if you multiply binomials – each factor has two terms.

Note the last two examples are “special cases” that you’ll see a lot: difference of two squares, and perfect square trinomials; note the shortcuts with these cases.

Multiplying Binomials FOIL FOILing Polynomials


This technique seems more popular these days, since it can apply to polynomials of any size.  Really, what we are doing when we are FOILing is using the distributive method to make sure every term (variable or number) is multiplied by every other one and then you add them all up.  We can also think of this as “pushing through” the terms to every other term.

Also called “double distributing”, this way of multiplying binomials is more popular now, since it can be used for any polynomial.

Let’s do a couple of the problems above and also see how “pushing through” can be used with polynomial products when we don’t have two binomials:

Multiplying Polynomials

More multiplying polynomials and also factoring polynomials, including by Completing the Square, can be found in the Introduction to Multiplying Polynomials,  Introduction to Quadratics and the Factoring Quadratics and Completing the Square sections.