Lesson 2.11

Sum and Difference of Cubes

You already know factors as . Cubes have their own special formulas — and you just need to memorize the SOAP pattern.

Introduction

The difference of squares is one of the first shortcuts you learn. Now we tackle cubes — expressions like or . Each factors into a binomial times a trinomial.

Past Knowledge

Difference of squares: .

Today's Goal

Factor sum and difference of cubes using the formulas and SOAP mnemonic.

Future Success

You already saw in long division. Now you get it instantly from the formula.

Key Concepts

The Two Formulas

Sum of Cubes

Difference of Cubes

The SOAP Mnemonic

The signs in the factored form follow the pattern S·O·A·P:

SSame sign as the original

OOpposite sign

AAlways

PPositive (the last term)

Common Perfect Cubes

125

Worked Examples

Example 1: Sum of Cubes

Basic

Factor .

Identify and

, . So , .

Apply the Formula (SOAP: +, −, +)

Example 2: Difference of Cubes

Intermediate

Factor .

Identify and

, . So , .

Apply the Formula (SOAP: −, +, +)

Example 3: GCF First, Then Cubes

Advanced

Factor completely.

Factor the GCF First

Now Apply the Cube Formula

, so , .

Common Pitfalls

Confusing with Difference of Squares

but gives a trinomial as the second factor. Don't try to use the squares pattern on cubes.

Forgetting to Square the Coefficients

If , then , NOT . The coefficient gets squared too!

Real-Life Applications

Volume formulas often involve cubes. If a storage container has volume cubic feet, factoring as lets you find the dimensions or solve for the exact size that gives zero wasted space.