# 4.7 Factoring the Sum and Difference of Cubes

**S**ame

**O**pposite

**A**lways

**P**ositive. For example, consider the following example.

*same*as the sign between [latex]x^3 - 2^3[/latex]. The sign of the [latex]2x[/latex] term is

*opposite*the sign between [latex]x^3 - 2^3[/latex]. And the sign of the last term, [latex]4[/latex], is

*always positive*.

## Sum and Difference of Cubes

We can factor the sum of two cubes as

[latex]a^3 + b^3 = (a + b)(a^2 - ab + b^2)[/latex]

We can factor the difference of two cubes as

[latex]a^3 - b^3 = (a - b)(a^2 + ab + b^2)[/latex]

## How To

**Given a sum of cubes or difference of cubes, factor it. **

- Confirm that the first and last term are cubes, [latex]a^3 + b^3[/latex] or [latex]a^3 - b^3[/latex]
- For a sum of cubes, write the factored form as [latex](a + b)(a^2 - ab + b^2)[/latex]. For a difference of cubes, write the factored form as [latex](a - b)(a^2 + ab + b^2)[/latex].

## Example 1: Factoring a Sum of Cubes

- Factor [latex]x^3 + 512[/latex].

*Analysis*

After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored further. However, the trinomial portion cannot be factored, so we do not need to check.

- Factor the sum of cubes: [latex]216a^3+b^3[/latex].

## Example 2: Factoring a Difference of Cubes

- Factor [latex]8x^3 - 125[/latex].

*Analysis*

Just as with the sum of cubes, we will not be able to further factor the trinomial portion.

- Factor the difference of cubes: [latex]1000x^3−1[/latex]

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