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The Sum of Cubes

Stephen Sacchetti, Joshua White
  • Author
    Stephen Sacchetti

    Stephen graduated from Haverford College with a B.S. in Mathematics in 2011. For the past ten years, he has been teaching high school math and coaching teachers on best practices. In 2015, Stephen earned an M.S. Ed from the University of Pennsylvania where he currently works as an adjunct professor.

  • Instructor
    Joshua White

    Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

In this lesson, learn what the sum of cubes is. Once this is understood, learn about how to factor sum of cubes and how it is done from sum of cubes examples. Updated: 10/08/2021

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.

The

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 x2 - 4 or 25x4 - 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:

a2 - b2 = (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, x3 - 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, y3 + 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:

a3 + b3 = (a + b) (a2 - ab + b2)

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

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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:

  • x3 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, 64z9 is a cube because there is an expression (4z3) that, when multiplied by itself three times (4z3)(4z3)(4z3), will equal 64z9.

Note that every part of each term must be a cube; 7x6 and 8y2 are not cubes because 7 is not a cube (although x6 is) and y2 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:

a3 + b3 = (a + b) (a2 - ab + b2)

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 (4y2). 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:
y5 + 27

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Video Transcript

Difference of Squares

You've probably come across factoring problems where an expression had two terms, such as x2 - 4 or 25x4 - 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:

a2 - b2 = (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, x3 - 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, y3 + 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:

a3 + b3 = (a + b) (a2 - ab + b2)

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:

  • x3 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, 64z9 is a cube because there is an expression (4z3) that, when multiplied by itself three times (4z3)(4z3)(4z3), will equal 64z9.

Note that every part of each term must be a cube; 7x6 and 8y2 are not cubes because 7 is not a cube (although x6 is) and y2 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:

a3 + b3 = (a + b) (a2 - ab + b2)

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 (4y2). 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:
y5 + 27

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Create your account

Frequently Asked Questions

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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