## 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

a^{2} - b^{2} = (a - b)(a + b)

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

a^{3} - b^{3} = (a - b)(a^{2} +ab + b^{2})

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

a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})

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.