## 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: $a^{3}+b^{3}$ and $a^{3}-b^{3}$.

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 $x$ can be represented by $x^{3}$. (Notice the exponent!)

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

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

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

### Example

Does $(a+b)(a^{2}–ab+b^{2})=a^{3}+b^{3}$?

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

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

### The Sum of Cubes

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

### Examples

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

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

### Example

Factor $x^{3}+8y^{3}$.

Now try another one.

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

### Example

Factor $16m^{3}+54n^{3}$.

## Difference of Cubes

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

### The Difference of Cubes

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

### Examples

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

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

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

Factored form of $a^{3}+b^{3}$: $\left(a+b\right)\left(a^{2}-ab+b^{2}\right)$

Factored form of $a^{3}-b^{3}$: $\left(a-b\right)\left(a^{2}+ab+b^{2}\right)$

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

“Remember one sequence of variables: $a^{3}b^{3}=\left(a\,b\right)\left(a^{2}ab\,b^{2}\right)$. There are $4$ 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 $+$.”

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

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

### Example

Factor $8x^{3}–1,000$.

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

### Example

Factor $r^{9}-8s^{6}$.

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

Always remember to factor out any common factors first.