Factoring a Difference or a Sum of Two Cubes

We can divide a³ - b³ by a - b and get the quotient a² + ab + b² and no remainder. So a - b is a factor of a³ - b³, a difference of two cubes. If you divide a³ + b³ by a + b, you will get the quotient a² - ab + b² and no remainder. Try it. So a + b is a factor of a³ + b³, a sum of two cubes. These results give us two more factoring rules.

 

Factoring a Difference or a Sum of Two Cubes

a³ - b³ = (a - b)(a² + ab + b²)

a³ + b³ = (a + b)(a² - ab + b²) 

 

Example 1

Factoring a difference or a sum of two cubes

Factor each polynomial.

a) x³ - 8

b) y³ + 1

c) 8z³ - 27

Solution

a) Because 8 = 2³, we can use the formula for factoring the difference of two cubes. In the formula a³ - b³ = (a - b)(a² + ab + b²) , let a = x and b = 2:

x³ - 8 = (x - 2)(x² + 2x + 4)