## Sum or Difference of Cubes

A **sum of cubes** or **difference of cubes** have two terms each of which are **perfect cubes**. That is, they can be **cube rooted** “nicely”.

1, 8, 27, 64, 125, ldots are perfect squares, as well as polynomials like x^3, y^3, x^6, y^9, a^3b^6, etc.

Unlike a difference of squares, there can be a plus or minus sign in between the two terms.

##### How to Factor a Sum or Difference of Squares

This is more easily explained using examples but there are some “lyrics to help you remember how to factor a sum or difference of cubes:

## Cube root the first term, keep the sign, cube root the second term

in order to make the first factor. Then, use the first factor to:

## Square the first term, negative product, square the last term.

Here are two basic examples: x^3+y^3=(x+y)(x^2-xy+y^2) x^3-y^3=(x-y)(x^2+xy+y^2)

For example: x^3-8

The cube root of the first term is x. The cube root of the second term is 2. Therefore:x^3-8=(x-2)(x^2+2x+4)

### Practice

Factor: 45x^2-500

###### SOLUTION

Using the Law of Division: 2^{11} div 2^8=2^{11-8}=2^3

Evaluating: 2^3=8

Factor: 25a^6-36b^4

###### SOLUTION

Using the Law of Powers: (m^0)^{10}=m^0

We should more properly use the Law of Zero Exponents to write: m^0=1

Factor: x^2+49

###### SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: x+4=7

Therefore: x=3

Factor: x^4-16

###### SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: x+4=7

Therefore: x=3

## No Comments