How Do You Factor Rational Fractions?

To factor rational fractions, you primarily focus on simplifying them by factoring both the numerator and the denominator and then canceling out any common factors. Here's a breakdown of the process, based on the reference provided:

Steps to Factoring Rational Fractions

Here's how to factor rational fractions, explained in clear steps:

1. Factor the Numerator and Denominator: The first step involves factoring the numerator and the denominator of the rational fraction as much as possible. This means breaking them down into their simplest multiplicative components.
2. Cancel Common Factors: Next, identify and cancel out any factors that appear in both the numerator and the denominator. This process simplifies the expression. According to the provided reference, "Cancel out any factors that are in both the numerator and denominator. The result is your simplified expression."

Example

Let's illustrate with an example:

Suppose you have the rational fraction:

(x² - 4) / (x² + 4x + 4)

• Factor the Numerator: The numerator (x² - 4) is a difference of squares, which factors to (x - 2)(x + 2).
• Factor the Denominator: The denominator (x² + 4x + 4) is a perfect square trinomial, which factors to (x + 2)(x + 2).

So, the expression becomes:

((x - 2)(x + 2)) / ((x + 2)(x + 2))

• Cancel Common Factors: Notice that (x + 2) appears in both the numerator and the denominator. Cancel one instance of (x + 2) from each.

This leaves you with:

(x - 2) / (x + 2)

This is the simplified form of the rational fraction.