Factoring the Difference of Cubes: A Step-by-Step Guide

In the realm of algebra, factoring is a crucial skill that allows us to break down complex expressions into simpler ones. One particular type of factoring that often arises is factoring the difference of cubes. This technique is applicable to expressions where we have two perfect cubes subtracted from each other. This guide will walk you through the process of factoring the difference of cubes step-by-step, making it a breeze to understand.

Understanding the Difference of Cubes

Before we delve into the steps, let's define what we mean by the difference of cubes. It's an expression that takes the form of:

a³ - b³

where 'a' and 'b' can be any real numbers or expressions.

The Difference of Cubes Formula

The key to factoring the difference of cubes lies in the following formula:

a³ - b³ = (a - b)(a² + ab + b²)

This formula tells us that the difference of cubes can be expressed as the product of two factors: a binomial (a - b) and a trinomial (a² + ab + b²).

Steps for Factoring the Difference of Cubes

Now, let's break down the process into clear steps:

1. Identify the perfect cubes: Begin by recognizing the terms that are perfect cubes. A perfect cube is a number or expression that can be obtained by cubing another number or expression. For example, 8 is a perfect cube because 8 = 2³. Similarly, x³ is a perfect cube because x³ = (x)³.
2. Apply the formula: Once you've identified the perfect cubes, substitute them into the difference of cubes formula (a³ - b³ = (a - b)(a² + ab + b²)).
3. Simplify: After applying the formula, simplify the expression by multiplying out the terms and combining like terms.

Example

Let's factor the expression x³ - 8.

1. Identify the perfect cubes: We have x³ and 8, which are perfect cubes because x³ = (x)³ and 8 = 2³.
2. Apply the formula: Substitute a = x and b = 2 into the formula: (x - 2)(x² + 2x + 2²).
3. Simplify: The simplified expression is (x - 2)(x² + 2x + 4).

Therefore, the factored form of x³ - 8 is (x - 2)(x² + 2x + 4).

Conclusion

Factoring the difference of cubes is a valuable technique in algebra that allows us to simplify expressions and solve equations. By following the steps outlined in this guide, you can master this skill and apply it to various mathematical problems. Remember to identify the perfect cubes, apply the formula, and simplify the resulting expression. With practice, factoring the difference of cubes will become second nature to you.