Factoring a Sum or Difference of Two Cubes

This product will lead us to another factoring pattern.

(a + b)(a² - ab + b²)

To find the product, multiply each term in (a + b) by each term in (a² - ab + b²).

= a · a² - a · ab + a · b² + b · a² - b · ab + b · b²

= a³ - a²b + ab² + a²b - ab² + b³

Combine like terms.

= a³ + b³

The four middle terms add to zero.

The result is a³ + b³, the sum of two cubes.

Note the structure of the two terms:

• The first term, a³, is a perfect cube.

• The last term, b³, is a perfect cube.

• The terms are added.

When we recognize this pattern, we can immediately factor the sum of two cubes as follows:

a³ + b³ = (a + b)(a² - ab + b²)

Note:

You may want to memorize a few perfect cubes.

1³ = 1

2³ = 8

3³ = 27

4³ = 64

5³ = 125

To check if a number is a perfect cube, use the  key on your calculator to see if the cube root is an integer.

Pattern — To Factor the Sum or Difference of Two Cubes

a³ + b³ = (a + b)(a² - ab + b²)

a³ - b³ = (a - b)(a² + ab + b²)

Example 1

Factor: y³ + 64

Solution

Step 1 Decide if the given polynomial fits a pattern.

The first term, y³, is a perfect cube, (y)³.

The last term, 64, is a perfect cube, (4)³.

The terms are added.

Therefore, y³ + 64 is a sum of two cubes.

The result is:

y³ + 64 = (y + 4)(y² - 4y + 16).

You can multiply to check the factorization. We leave the check to you.

Example 2

Factor: 8w³ - 1

Solution

Step 1 Decide if the given polynomial fits a pattern.

The first term, 8w³, is a perfect cube, (2w)³.

The last term, 1, is a perfect cube, (1)³.

The terms are subtracted.

Therefore, 8w³ - 1 is a difference of two cubes.

Step 2 Identify a and b. Then substitute in the pattern and simplify.

In the factoring pattern for a difference of two cubes, substitute 2w for a and 1 for b.

You can multiply to check the factorization. We leave the check to you.