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

In the realm of algebra, factoring is a fundamental skill that allows us to manipulate expressions and solve equations. One specific type of factoring involves recognizing and manipulating expressions known as the difference of cubes. This blog post will guide you through the process of factoring the difference of cubes, providing a clear and concise understanding of the steps involved.

What is the Difference of Cubes?

The difference of cubes refers to a binomial expression of the form a³ - b³, where 'a' and 'b' represent any real numbers or algebraic expressions. The key characteristic of this expression is that both terms are perfect cubes, meaning they can be expressed as the cube of another number or expression.

The Difference of Cubes Formula

The difference of cubes formula provides a straightforward way to factor expressions of this type. It states:

a³ - b³ = (a - b)(a² + ab + b²)

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

Steps for Factoring the Difference of Cubes

To factor the difference of cubes, follow these steps:

1. Identify the Perfect Cubes: Determine whether both terms in the binomial expression are perfect cubes. If they are, identify the cube roots of each term.
2. Apply the Formula: Substitute the cube roots into the difference of cubes formula. The first term in the binomial factor will be the difference of the cube roots, and the trinomial factor will have the squares of the cube roots and their product.
3. Simplify: Simplify the resulting expression by combining like terms, if possible.

Example: Factoring 8x³ - 27

Let's factor the expression 8x³ - 27:

1. Identify the Perfect Cubes: 8x³ is the cube of 2x (2x)³ = 8x³, and 27 is the cube of 3 (3)³ = 27.
2. Apply the Formula: Using the difference of cubes formula, we get: (2x)³ - (3)³ = (2x - 3)( (2x)² + (2x)(3) + (3)²)
3. Simplify: Simplifying the expression, we get: (2x - 3)(4x² + 6x + 9)

Therefore, the factored form of 8x³ - 27 is (2x - 3)(4x² + 6x + 9).

Practice Problems

Here are some practice problems for you to try:

1. Factor x³ - 125
2. Factor 64y³ - 1
3. Factor 27z³ - 8

Conclusion

Factoring the difference of cubes is a valuable skill in algebra. By understanding the formula and following the steps outlined in this guide, you can confidently factor expressions of this type. Remember to practice regularly to master this technique and enhance your algebraic proficiency.