# The Sum of Cubes

## Sum of Cubes

A **sum of cubes** is the sum of two expressions, each of which is a **perfect cube**. For example, the expression {eq}x^3 + 8 {/eq} is a sum of cubes since {eq}x^3 {/eq} is a perfect cube and 8 is a perfect cube ({eq}8 = 2^3 {/eq}). A more complicated expression such as {eq}64x^3 + 27y^3 {/eq} is also a sum of cubes since {eq}64x^3 = (4x)^3 {/eq} and {eq}27y^3 = (3y)^3 {/eq}. Please note, that expressions such as {eq}2x^3 {/eq} are not perfect cubes. Even though the x term is cubed, the coefficient is not a perfect cube and this expression cannot be rewritten as {eq}(ax)^3 {/eq} where a is a whole number.

### Factoring Sum of Cubes

Before jumping into how to factor a sum of cubes, it is important to know a few of the perfect cubes. Just as the perfect squares are well known up to 100 (and possibly up to 144), so too is it helpful to know the perfect cubes up to {eq}10^3 {/eq} and maybe even {eq}12^3 {/eq}. Study the table before continuing onto factoring through a sum of cubes formula:

x | x^3 |
---|---|

1 | 1 |

2 | 8 |

3 | 27 |

4 | 64 |

5 | 125 |

6 | 216 |

7 | 343 |

8 | 512 |

9 | 729 |

10 | 1000 |

11 | 1331 |

12 | 1728 |

Armed with this knowledge, it will be easier to recognize perfect cubes and, as a result, to factor cubes as well. In particular, when given the sum of two cubes in the form {eq}a^3 + b^3 {/eq}, the expression can be factored as shown:

#### {eq}a^3 + b^3 = (a + b)(a^2 - ab + b^2) {/eq}

It is important, therefore, to ensure that the expression is simplified to be a sum of two cubes before using the sum of cubes formula. For example, in an expression such as {eq}8x^3 + 27 {/eq}, the first term must be simplified as {eq}8x^3 = (2x)^3 {/eq} and so {eq}a = 2x {/eq}. Now, rewrite the expression as the sum of two cubes:

{eq}8x^3 + 27 = (2x)^3 + 3^3 \\ 8x^3 + 27 = (2x + 3)((2x)^2 - 2x\cdot 3 + 3^2) = (2x + 3)(4x^2 - 6x + 9) {/eq}

One of these factors can also be seen graphically. Noticing that the factor {eq}(2x + 3) {/eq} exists means that this expression equals zero when {eq}x=-\frac{3}{2} {/eq}. This can be confirmed by graphing {eq}8x^3 + 27 {/eq} as shown below.

### Sum of Cubes Examples

In this section, sum of cubes examples will be given that increase in difficulty. Follow along with the work shown and refer back to the table of perfect cubes as needed.

{eq}\ \\ \ {/eq}

#### Example 1: Use the sum of cubes formula to factor {eq}x^3 + 125 {/eq}

In this case, recognize that {eq}125=5^3 {/eq}, thus this expression is already written as the sum of two cubes. Factoring by using the sum of cubes formula then gives:

{eq}x^3 + 125 = x^3 + 5^3 {/eq}

## Difference of Squares

You've probably come across factoring problems where an expression had two terms, such as *x*2 - 4 or 25*x*4 - 16.

If both terms were squares and had opposite signs (i.e., one term was positive and one term was negative), then you could factor it as a difference of squares using the difference of squares formula:

*a*2 - *b*2 = (*a* + *b*) (*a* - *b*)

But what should you do when you come across a two-term expression where the terms each have the same sign? Or if the terms are not squares but are instead cubes?

If both terms are cubes, then it may be possible to factor the expression as either a difference of cubes or a sum of cubes, depending on the signs of the terms. An expression with opposite signs (for example, *x*3 - 8) could be a difference of cubes, which is covered in a separate lesson. An expression where both terms have the same sign (for example, *y*3 + 1), either both positive or both negative, could be factored as a sum of cubes, which is the focus of this lesson.

A **sum of cubes** is a two-term expression where both terms are cubes and each term has the same sign. It is factored according to the following formula:

*a*3 + *b*3 = (*a* + *b*) (*a*2 - *ab* + *b*2)

How can you determine if an expression can be factored as a sum of cubes?

Here, a = x and b = 5, so recall the formula {eq}a^3 + b^3 = (a + b)(a^2 - ab + b^2) {/eq} and make the substitution to find:

{eq}x^3 + 125 = (x + 5)(x^2 - 5x + 25) {/eq}

{eq}\ \\ \ {/eq}

#### Example 2: Use the sum of cubes formula to factor {eq}216x^3 + 1 {/eq}

In this case, notice that {eq}216x^3 = (6x)^3 {/eq}, so

{eq}216x^3 + 1 = (6x)^3 + 1^3 = (6x + 1)(36x^2 - 6x + 1) {/eq}

{eq}\ \\ \ {/eq}

#### Example 3: Factor the expression {eq}512x^3 + 343y^3 {/eq}

In this case, recall that {eq}512 = 8^3 {/eq} and {eq}343 = 7^3 {/eq}, thus the expression becomes:

{eq}512x^3 + 343y^3 = (8x)^3 + (7y)^3 {/eq}

and can be factored as:

{eq}512x^3 + 343y^3 = (8x + 7y)(64x^2 - 56xy + 49y^2) {/eq}

{eq}\ \\ \ {/eq}

#### Example 4: Factor the expression {eq}x^6 + 64y^6 {/eq}

In this case, notice that neither term has an exponent of 3. However, each term can be rewritten using laws of exponents to say {eq}x^6 = (x^2)^3 {/eq} and {eq}64y^6 = (4y^2)^3 {/eq}. As a result, set {eq}a = x^2 {/eq} and {eq}b = 4y^2 {/eq}. This gives the following factorization:

{eq}x^6 + 64y^6 = (x^2 + 4y^2)(x^4 - 4x^2y^2 + 16y^4) {/eq}

{eq}\ \\ \ {/eq}

#### Example 5: Factor the expression {eq}729x^3 + 144y^3 {/eq}

In this case, recall that {eq}729 = 9^3 {/eq}, but notice that 144 is not a perfect cube. Therefore this expression cannot be factored using the sum of cubes formula.

## Lesson Summary

A **sum of cubes** is an expression where two perfect cubes are added together. Algebraically, this looks like {eq}a^3 + b^3 {/eq} and can be factored as

#### {eq}a^3 + b^3 = (a + b)(a^2 - ab + b^2) {/eq}

It is important to rewrite any expressions involving cubes as the sum of two cubed quantities before using this formula, so be sure to rewrite any terms with coefficients in a factored form. Knowing the perfect cubes will help the factoring process go more smoothly as will recalling the idea of exponent properties (namely that exponents that are multiples of 3 create **perfect cubes**).

## Is an Expression a Sum of Cubes?

An expression must meet two criteria in order to be factored as a sum of cubes. First, each term must be a cube. In other words, each term must be the result of multiplying the same expression by itself three times. Here are some examples:

*x*3 is a cube because it is a result of*x*multiplied by itself three times

(*x***x***x*).- 27 is a cube because it is the result of 3 multiplied by itself three times

(3 * 3 * 3).

Additionally, you may find a cube that contains both numbers and variables. For example, 64*z*9 is a cube because there is an expression (4*z*3) that, when multiplied by itself three times (4*z*3)(4*z*3)(4*z*3), will equal 64*z*9.

Note that every part of each term must be a cube; 7*x*6 and 8*y*2 are not cubes because 7 is not a cube (although *x*6 is) and *y*2 is not a cube (although 8 is).

Second, each term must have the same sign, usually both positive. Note that if both signs are negative, you can factor a -1 out of both terms to make them each positive. If both terms have opposite signs, then you may want to try and factor the expression as a difference of squares or a difference of cubes. Now, let's see how you use the sum of cubes formula to factor a problem.

## Factoring a Sum of Cubes

For a sum of cubes, you'll use the formula already mentioned:

*a*3 + *b*3 = (*a* + *b*) (*a*2 - *ab* + *b*2)

Note that *a* and *b* represent the individual expressions that are cubed. They could each be a variable (*x*), a number (3) or some combination of both (4*y*2). First, you must determine what *a* and *b* are. Essentially you're asking, what do I cube to get the first term and what do I cube to make the second term? After you've done that, you will plug in the expressions you found for *a* and *b* into the formula and simplify them to finish the factoring. Let's see some examples.

## Examples

Can the following expressions be factored as a sum of cubes? If yes, factor.

Example 1: *y*5 + 27

This expression cannot be factored as a sum of cubes because the first term (*y*5) is not a cube. In other words, there is nothing that can be multiplied by itself three times to equal *y*5. Therefore, the expression cannot be factored.

Example 2:

*x*3 + 64

This expression can be factored as a sum of cubes since both terms have the same sign (+) and each expression is a cube.

*x* can be cubed to give *x*3

4 can be cubed to make 64.

Thus, plugging *a* = *x* and *b* = 4 into the sum of cubes formula gives:

(*x* + 4) (x2 - (*x*)(4) + 42)

Simplifying (*x*)(4) to 4*x* and 42 to 16 gives us the final answer of

(*x* + 4) (*x*2 - 4*x* + 16).

Example 3:

8*x*3 + 27*y*6

The two terms both have the same sign, but is each term a cube? Yes.

(2*x*)(2*x*)(2*x*) = 8*x*3 and

(3*y*2)(3*y*2)(3y2) = 27*y*6

Thus, *a* = 2*x* and *b* = 3*y*2. Plugging these into the formula gives the following factorization:

(2*x*)3 + (3*y*2)3

= (2*x* + 3*y*2)[(2*x*)2 - (2*x*)(3*y*2) + (3*y*2)2]

=(2*x* + 3*y*2)(4*x*2 - 6*xy*2 + 9*y*4

Example 4:

-2*x*3 - 54

First, notice that while the terms each have the same sign (-), they are not both positive like in the previous examples. That's not a problem though, since we pull out a factor of a negative number to make each term positive.

Additionally, see that -2 and -54 are not cubes. However, they do have a factor of 2 in common or, in this case, technically -2. First, you will factor a -2 out of both terms to give:

-2(*x*3 + 27)

Then you can proceed to factor *x*3 + 27 as a sum of cubes where *a* = *x* and *b* = 3. Thus, we have:

-2(*x* + 3) (*x*2 - (*x*)(3) + 32)

= -2 (*x* + 3) (*x*2 - 3*x* + 9)

## Lesson Summary

A **sum of cubes** is a two-term expression where both terms are cubes and each term has the same sign. It is factored according to the following formula:

*a*3 + *b*3 = (*a* + *b*) (*a*2 - *ab* + *b*2)

In order to factor an expression as a sum of cubes, you must first check to see if it meets the criteria:

- The expression must have two terms, each with the same sign.
- Each term must be a cube.

Remember that sometimes you may need to pull out a common factor first before you can determine if the terms are cubes. If those criteria are met, then you can factor the sum of cubes according to the formula. Finally, make sure you simplify your answer after plugging the values into the formula.

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## Difference of Squares

You've probably come across factoring problems where an expression had two terms, such as *x*2 - 4 or 25*x*4 - 16.

If both terms were squares and had opposite signs (i.e., one term was positive and one term was negative), then you could factor it as a difference of squares using the difference of squares formula:

*a*2 - *b*2 = (*a* + *b*) (*a* - *b*)

But what should you do when you come across a two-term expression where the terms each have the same sign? Or if the terms are not squares but are instead cubes?

If both terms are cubes, then it may be possible to factor the expression as either a difference of cubes or a sum of cubes, depending on the signs of the terms. An expression with opposite signs (for example, *x*3 - 8) could be a difference of cubes, which is covered in a separate lesson. An expression where both terms have the same sign (for example, *y*3 + 1), either both positive or both negative, could be factored as a sum of cubes, which is the focus of this lesson.

A **sum of cubes** is a two-term expression where both terms are cubes and each term has the same sign. It is factored according to the following formula:

*a*3 + *b*3 = (*a* + *b*) (*a*2 - *ab* + *b*2)

How can you determine if an expression can be factored as a sum of cubes?

## Is an Expression a Sum of Cubes?

An expression must meet two criteria in order to be factored as a sum of cubes. First, each term must be a cube. In other words, each term must be the result of multiplying the same expression by itself three times. Here are some examples:

*x*3 is a cube because it is a result of*x*multiplied by itself three times

(*x***x***x*).- 27 is a cube because it is the result of 3 multiplied by itself three times

(3 * 3 * 3).

Additionally, you may find a cube that contains both numbers and variables. For example, 64*z*9 is a cube because there is an expression (4*z*3) that, when multiplied by itself three times (4*z*3)(4*z*3)(4*z*3), will equal 64*z*9.

Note that every part of each term must be a cube; 7*x*6 and 8*y*2 are not cubes because 7 is not a cube (although *x*6 is) and *y*2 is not a cube (although 8 is).

Second, each term must have the same sign, usually both positive. Note that if both signs are negative, you can factor a -1 out of both terms to make them each positive. If both terms have opposite signs, then you may want to try and factor the expression as a difference of squares or a difference of cubes. Now, let's see how you use the sum of cubes formula to factor a problem.

## Factoring a Sum of Cubes

For a sum of cubes, you'll use the formula already mentioned:

*a*3 + *b*3 = (*a* + *b*) (*a*2 - *ab* + *b*2)

Note that *a* and *b* represent the individual expressions that are cubed. They could each be a variable (*x*), a number (3) or some combination of both (4*y*2). First, you must determine what *a* and *b* are. Essentially you're asking, what do I cube to get the first term and what do I cube to make the second term? After you've done that, you will plug in the expressions you found for *a* and *b* into the formula and simplify them to finish the factoring. Let's see some examples.

## Examples

Can the following expressions be factored as a sum of cubes? If yes, factor.

Example 1: *y*5 + 27

This expression cannot be factored as a sum of cubes because the first term (*y*5) is not a cube. In other words, there is nothing that can be multiplied by itself three times to equal *y*5. Therefore, the expression cannot be factored.

Example 2:

*x*3 + 64

This expression can be factored as a sum of cubes since both terms have the same sign (+) and each expression is a cube.

*x* can be cubed to give *x*3

4 can be cubed to make 64.

Thus, plugging *a* = *x* and *b* = 4 into the sum of cubes formula gives:

(*x* + 4) (x2 - (*x*)(4) + 42)

Simplifying (*x*)(4) to 4*x* and 42 to 16 gives us the final answer of

(*x* + 4) (*x*2 - 4*x* + 16).

Example 3:

8*x*3 + 27*y*6

The two terms both have the same sign, but is each term a cube? Yes.

(2*x*)(2*x*)(2*x*) = 8*x*3 and

(3*y*2)(3*y*2)(3y2) = 27*y*6

Thus, *a* = 2*x* and *b* = 3*y*2. Plugging these into the formula gives the following factorization:

(2*x*)3 + (3*y*2)3

= (2*x* + 3*y*2)[(2*x*)2 - (2*x*)(3*y*2) + (3*y*2)2]

=(2*x* + 3*y*2)(4*x*2 - 6*xy*2 + 9*y*4

Example 4:

-2*x*3 - 54

First, notice that while the terms each have the same sign (-), they are not both positive like in the previous examples. That's not a problem though, since we pull out a factor of a negative number to make each term positive.

Additionally, see that -2 and -54 are not cubes. However, they do have a factor of 2 in common or, in this case, technically -2. First, you will factor a -2 out of both terms to give:

-2(*x*3 + 27)

Then you can proceed to factor *x*3 + 27 as a sum of cubes where *a* = *x* and *b* = 3. Thus, we have:

-2(*x* + 3) (*x*2 - (*x*)(3) + 32)

= -2 (*x* + 3) (*x*2 - 3*x* + 9)

## Lesson Summary

**sum of cubes** is a two-term expression where both terms are cubes and each term has the same sign. It is factored according to the following formula:

*a*3 + *b*3 = (*a* + *b*) (*a*2 - *ab* + *b*2)

In order to factor an expression as a sum of cubes, you must first check to see if it meets the criteria:

- The expression must have two terms, each with the same sign.
- Each term must be a cube.

Remember that sometimes you may need to pull out a common factor first before you can determine if the terms are cubes. If those criteria are met, then you can factor the sum of cubes according to the formula. Finally, make sure you simplify your answer after plugging the values into the formula.

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#### What is the sum of cubes formula?

The sum of cubes formula gives a way to factor an expression written in the form a^3 + b^3. In particular, it says a^3 + b^3 = (a + b)(a^2 - ab +b^2).

#### How do you factor the sum of two cubes?

First, ensure that the expression is written as a^3 + b^3. Then from there, use the formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

#### What is the first step in factoring the sum of two cubes?

When factoring the sum of two cubes, first identify which quantities are being cubed. For instance, if given the term 64x^3, rewrite it as (4x)^3.

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