How do you factor a binomial?

A binomial is an expression with two terms separated by either addition or subtraction. The goal is to make it all one term — with everything multiplied together. This is accomplished by factoring the two terms. You can use four basic methods to factor a binomial. If none of these methods works, the expression is considered to be prime — meaning it cannot be factored.

The rules or patterns to use when doing the factoring are as follows:

Rule 1: Factoring out the Greatest Common Factor

ab + ac = a(b + c)

Rule 2: Factoring using the pattern for the differences of squares

a2 - b2 = (a - b)(a + b)

Rule 3: Factoring using the pattern for the difference of cubes

a3 - b3 = (a - b)(a2 +ab + b2)

Rule 4: Factoring using the pattern for the sum of cubes

a3 + b3 = (a + b)(a2 - ab + b2)

The challenge is in determining which factoring method to use. If you recognize that both terms are perfect squares and they're subtracted, then Rule 2 makes sense. If both terms are perfect cubes, then Rule 3 or 4 will work. If they have one or more factors in common, then use Rule 1. Sometimes, you get to use more than one rule to complete the job.