Factoring the Sum of Cubes: A Simple Guide

Factoring algebraic expressions is a fundamental skill in algebra. It involves breaking down a complex expression into simpler ones, which are easier to manipulate and solve. One specific type of factoring involves the sum of cubes. Understanding this concept can help you solve various mathematical problems.

What is the Sum of Cubes?

The sum of cubes refers to a binomial expression where both terms are perfect cubes. A perfect cube is a number or variable that can be obtained by cubing an integer or variable. For example, 8 is a perfect cube because it’s the result of 2 cubed (2 x 2 x 2 = 8). Similarly, x³ is a perfect cube because it’s obtained by cubing x (x x x = x³).

The general form of the sum of cubes is:

a³ + b³

Factoring the Sum of Cubes

Factoring the sum of cubes follows a specific formula:

a³ + b³ = (a + b)(a² – ab + b²)

Let’s break down the formula:

• (a + b): This is the sum of the cube roots of the original terms.
• (a² – ab + b²): This is a trinomial that results from squaring the first term (a²), multiplying the first and second terms (ab) and changing the sign, and squaring the second term (b²).

Step-by-Step Guide to Factoring

Here’s how to factor the sum of cubes in a few simple steps:

1. **Identify the terms as perfect cubes:** Determine if both terms in the binomial expression are perfect cubes.
2. **Apply the formula:** Use the sum of cubes formula to factor the expression.
3. **Simplify if possible:** In some cases, the resulting trinomial might be further factorable. If so, simplify the expression.

Example

Let’s factor the expression x³ + 8:

1. **Identify the perfect cubes:** x³ is a perfect cube of x, and 8 is a perfect cube of 2 (2 x 2 x 2 = 8).
2. **Apply the formula:** Using the formula, we get: (x + 2)(x² – 2x + 4).
3. **Simplify:** The trinomial (x² – 2x + 4) cannot be factored further.

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

Conclusion

Factoring the sum of cubes is a valuable skill in algebra. By understanding the formula and following a few simple steps, you can easily factor such expressions. This skill is applicable in various mathematical problems, including solving equations, simplifying expressions, and performing other algebraic operations.