Special Cases – Cubes

Learning Outcomes

  • Factor special products

Some interesting patterns arise when you are working with cubed quantities within polynomials. Specifically, there are two more special cases to consider: [latex]a^{3}+b^{3}[/latex] and [latex]a^{3}-b^{3}[/latex].

Let us take a look at how to factor sums and differences of cubes.

Sum of Cubes

The term “cubed” is used to describe a number raised to the third power. In geometry, a cube is a six-sided shape with equal width, length, and height; since all these measures are equal, the volume of a cube with width [latex]x[/latex] can be represented by [latex]x^{3}[/latex]. (Notice the exponent!)

Cubed numbers get large very quickly: [latex]1^{3}=1[/latex], [latex]2^{3}=8[/latex], [latex]3^{3}=27[/latex], [latex]4^{3}=64[/latex], and [latex]5^{3}=125[/latex]

Before looking at factoring a sum of two cubes, let us look at the possible factors.

It turns out that [latex]a^{3}+b^{3}[/latex] can actually be factored as [latex]\left(a+b\right)\left(a^{2}–ab+b^{2}\right)[/latex]. Check these factors by multiplying.


Does [latex](a+b)(a^{2}–ab+b^{2})=a^{3}+b^{3}[/latex]?

Did you see that? Four of the terms cancelled out, leaving us with the (seemingly) simple binomial [latex]a^{3}+b^{3}[/latex]. So, the factors are correct.

You can use this pattern to factor binomials in the form [latex]a^{3}+b^{3}[/latex], otherwise known as “the sum of cubes.”

The Sum of Cubes

A binomial in the form [latex]a^{3}+b^{3}[/latex] can be factored as [latex]\left(a+b\right)\left(a^{2}–ab+b^{2}\right)[/latex].


The factored form of [latex]x^{3}+64[/latex] is [latex]\left(x+4\right)\left(x^{2}–4x+16\right)[/latex].

The factored form of [latex]8x^{3}+y^{3}[/latex] is [latex]\left(2x+y\right)\left(4x^{2}–2xy+y^{2}\right)[/latex].


Factor [latex]x^{3}+8y^{3}[/latex].

Now try another one.

You should always look for a common factor before you follow any of the patterns for factoring.


Factor [latex]16m^{3}+54n^{3}[/latex].

Difference of Cubes

Having seen how binomials in the form [latex]a^{3}+b^{3}[/latex] can be factored, it should not come as a surprise that binomials in the form [latex]a^{3}-b^{3}[/latex] can be factored in a similar way.

The Difference of Cubes

A binomial in the form [latex]a^{3}–b^{3}[/latex] can be factored as [latex]\left(a-b\right)\left(a^{2}+ab+b^{2}\right)[/latex].


The factored form of [latex]x^{3}–64[/latex] is [latex]\left(x–4\right)\left(x^{2}+4x+16\right)[/latex].

The factored form of [latex]27x^{3}–8y^{3}[/latex] is [latex]\left(3x–2y\left)\right(9x^{2}+6xy+4y^{2}\right)[/latex].

Notice that the basic construction of the factorization is the same as it is for the sum of cubes; the difference is in the [latex]+[/latex] and [latex]–[/latex] signs. Take a moment to compare the factored form of [latex]a^{3}+b^{3}[/latex] with the factored form of [latex]a^{3}-b^{3}[/latex].

Factored form of [latex]a^{3}+b^{3}[/latex]: [latex]\left(a+b\right)\left(a^{2}-ab+b^{2}\right)[/latex]

Factored form of [latex]a^{3}-b^{3}[/latex]: [latex]\left(a-b\right)\left(a^{2}+ab+b^{2}\right)[/latex]

This can be tricky to remember because of the different signs. The factored form of [latex]a^{3}+b^{3}[/latex] contains a negative, and the factored form of [latex]a^{3}-b^{3}[/latex] contains a positive! Some people remember the different forms like this:

“Remember one sequence of variables: [latex]a^{3}b^{3}=\left(a\,b\right)\left(a^{2}ab\,b^{2}\right)[/latex]. There are [latex]4[/latex] missing signs. Whatever the first sign is, it is also the second sign. The third sign is the opposite, and the fourth sign is always [latex]+[/latex].”

Try this for yourself. If the first sign is [latex]+[/latex], as in [latex]a^{3}+b^{3}[/latex], according to this strategy, how do you fill in the rest: [latex]\left(a\,b\right)\left(a^{2}ab\,b^{2}\right)[/latex]? Does this method help you remember the factored form of [latex]a^{3}+b^{3}[/latex] and [latex]a^{3}–b^{3}[/latex]?

Let us go ahead and look at a couple of examples. Remember to factor out all common factors first.


Factor [latex]8x^{3}–1,000[/latex].

Here is one more example. Note that [latex]r^{9}=\left(r^{3}\right)^{3}[/latex] and that [latex]8s^{6}=\left(2s^{2}\right)^{3}[/latex].


Factor [latex]r^{9}-8s^{6}[/latex].

In the following two video examples, we show more binomials that can be factored as a sum or difference of cubes.

You encounter some interesting patterns when factoring. Two special cases—the sum of cubes and the difference of cubes—can help you factor some binomials that have a degree of three (or higher, in some cases). The special cases are:

  • A binomial in the form [latex]a^{3}+b^{3}[/latex] can be factored as [latex]\left(a+b\right)\left(a^{2}–ab+b^{2}\right)[/latex]
  • A binomial in the form [latex]a^{3}-b^{3}[/latex] can be factored as [latex]\left(a-b\right)\left(a^{2}+ab+b^{2}\right)[/latex]

Always remember to factor out any common factors first.