The Process

Factoring completely is a three step process:

1. Factor a GCF from the expression, if possible.
2. Factor a Trinomial, if possible.
3. Factor a Difference Between Two Squares as many times as possible.

First Example

Let's see how this applies to our initial example:

(x⁴ - 1)

Step 1

Step one is to factor a GCF. Since the GCF of x⁴ and 1 is 1, we skip this step.

Step 2

Since the expression only has two terms, we cannot factor a trinomial.

Step 3

Factoring (x⁴ - 1) as a difference between two squares results in

(x² + 1)(x² - 1).

Now be sure to remember the key phrase "as many times as possible." We must now look to see if there is anywhere else we can factor another Difference Between Two Squares. In (x² + 1), both terms are positive, so this cannot be factored. However, in (x² - 1), the second term is negative, and both terms are perfect squares otherwise. So (x² - 1) factors into (x + 1)(x - 1). As a result, our example expression is finally factored into

(x² + 1)(x + 1)(x - 1)

which is factored completely.

How did this differ from our first (and failed) attempt to factor the example? When Factoring Completely, we used the Difference Between Two Squares method more than once.

Proceed to the next page for another example.