In Factoring, Level 6 By Mario Salvador

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

EXAMPLE 1

Factor: 45x^2-500

SOLUTION

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

Evaluating: 2^3=8

EXAMPLE 2

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

EXAMPLE 3

Factor: x^2+49

SOLUTION

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

Therefore: x=3

EXAMPLE 4

Factor: x^4-16

SOLUTION

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

Therefore: x=3

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