# Factoring Sum and Difference of Two Perfect Cubes

We saw how to factor difference of two perfect squares using the following formula:

In this lesson, we will try to identify how to factor the difference and sum of two perfect cubes using a very specific pattern.

But first, let's get to know what a cube is.

A cube is a number that can be converted to a power raised to the exponent of 3.

List of first 11 perfect cube numerical values:

Algebraic expressions for perfect cubes:

Note that each power with an exponent of multiples of 3 is also a perfect cube number.

The formula in which the sum and difference of perfect squares are factored:

### Note:

Let and represent two real numbers.

Factoring the sum of two perfect cubes:

Factoring the difference of two perfect cubes:

There is a simple way to help you remember these patterns. Note carefully in the formula factored, the first binomial retains the same sign as between the first and second terms of the left-hand side. While a trinomial, we always put between the first term and the second term the opposite of the sign as between the first and second terms of the left-hand side. Finally, the third term is always preceded by a + sign.

Now let's take a variety of illustrations to factor out the addition and difference of two perfect cubes.

## Examples of Factoring Sum and Difference of Two Perfect Cubes.

#### Example1:

Factor the difference:

#### Solution:

This algebraic expression is written as a difference. We notice that the first number is raised to the power of 3. The number 27 is not. So, to factor this algebraic expression, it is necessary to convert the number 27 to a power of 3. What is this power then?

No doubt you know the answer, the number , if we decompose the number 9 we get , so

Now we get the difference of two perfect cubes. Thus, the expression can be factored using the second form in the rule above.

Factor the sum:

#### Solution:

In the same way, this algebraic expression is written as a sum. We notice that the second number is raised to the power of 3. The number 125 is not. So, to factor this algebraic expression, it is necessary to convert the number 125 to a power of 3. What is this power then?

The number , if we decompose the number 25 we get , so

Now we get the sum of two perfect cubes. Thus, the expression can be factored using the first form in the rule above.

Factor the sum:

#### Solution:

Rewrite this expression to obtain the sum of two perfect powers. Note that the number 8 is a perfect cube.

Now we get the sum of two perfect cubes. Thus, the expression can be factored using the first form in the rule above.

#### Example4:

Factor the difference:

#### Solution:

Rewrite this expression to obtain the difference of two perfect powers. Note that the two numbers 64 and 216 are a perfect cubes.

and

Now we get the difference of two perfect cubes. Thus, the expression can be factored using the second form in the rule above.

But the the expression obtained is not fully factored. Using factoring out the Greatest Common Factor:

• In the binomial, , Thus,
• In the trinomial, , Thus,

Therefore,

Factor the sum:

#### Solution:

Rewrite this expression to obtain the sum of two perfect powers. Note that the a perfect cubes.

Thus,

Factor the sum:

#### Solution:

This expression is written as a sum of two perfect cubes. The two terms are composite terms.