Up: Templates

Multiplying Binomials

(This page illustrates WebEQ Math Viewer scripting capabilities)

Many physical problems can be described by quadratic equations like images/demo1.png , and as a result, finding solutions to such equations is an important task. One of the main algebraic techniques for finding solutions is to factor the quadratic polynomial, which means writing the quadratic polynomial as a product of two first order terms or binomials. So the place to start with solving quadratic equations is to learn about binomials, and how to multiply them together.

Binomials

A binomial is a particularly simple kind of polynomial -- namely, one involving only an 'x' term and a constant term. We say 'x' here, but obviously the name of the variable doesn't matter. Some examples are:

In general, the form of a binomial is

where a and b are numbers. While the first example shows it is fine for a to be 1 (which makes it look like it isn't there until you think about it) and either a or b can be negative, we don't allow a or b to be zero. If we allowed a or b to be 0, we would only be left with one term, and at the moment, we are interested in binomials -- those with two terms.

Multiplying Binomials by the Book

Let's look at a concrete binomial multiplication problem. How do you multiply

out? The most direct approach is to use the basic laws of algebra and arithmetic, namely addition and the distributive law.

[Click the the "Next Step" button to go through the process. If you are unsure of the justification for each step, put the mouse over the equation and look for the reason in the status line at the bottom of your browser.]

The FIOL Shortcut

If you spend any amount of time multiplying binomials together, you will soon realize that you can cut out some of the steps from the procedure illustrated above. Many people use a mnemonic phrase "FIOL" to jump straight to the third step above:

Note that we wrote down the middle two terms (the 3x and the -2x) in the reverse order. The reason for this will be clear in a minute.

FIOL stands for "First, Inner, Outer, Last". The idea is to get the four terms on the right, by picking the right combinations of terms to multiply from the expression on the left. In general, you take one term from the first parenthesized expression and multiply it by a term from the second parenthesized expression. "First, inner, outer, last" reminds you which terms to take as illustrated below.

[Click the the "Next Step" button to watch how the FIOL shortcut works. Move mouse over the equations if you need a hint about what you are seeing.]

Check your Understanding

To make sure you have the hang of multiplying monomials, you might want to try your hand at the following problems:

[Click on the equations to reveal the answer]

WebEQ 2.5 Documentation Table of Contents