Factoring Completely

Introduction

Previous factoring lessons each focused on factoring a plynomial using a single pattern such as

Greatest Common Factor
Example:   3x² + 9x³ + 12x⁴   factored into   3x²(1 + 3x + 4x²)

Difference Between Two Squares
Example:   y² - 9  factored into  (y + 3)(y - 3)

Trinomial
Example:   x² - 2x - 3   factored into   (x + 1)(x - 3)

The lessons linked above give systematic techniques to factor certain types of polynomials. In practice, solving equations using factoring often requires the use of a more complex process called "Factoring Completely". This lesson explains how to factor completely by combining the three basic techniques listed above.

First, lets take a closer look at why we need the Factoring Completely process. Examine the expression below:

(x² + 1)(x + 1)(x - 1)

If we simplify this expression, we get:

(x² + 1)(x² + x - x - 1)
(x² + 1)(x² - 1)
(x⁴ + x² - x² - 1)
x⁴ - 1

Now you should recognize this expression as a difference between two squares. Using the technique presented in the Difference Between Two Squares lesson, we can factor this into

(x² + 1)(x² - 1)

But notice that this expression is not the same as the factored expression that we started with. We need another step to factor this into the expression that we started with. This is where Factoring Completely comes in.