3.7 Factoring the Sum or Difference of Cubes and a General Approach to Factoring

Learning Objectives

  • Factor the sum or difference of cubes.
  • Apply factoring strategies to completely factor polynomial expressions.

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 of cubes [latex] a^3+b^3 [/latex] and differences of cubes [latex] a^3-b^3 [/latex].

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 we talk about factoring sum of cubes, let’s review polynomial multiplication. What do we get when we multiply [latex]\left(a+b\right)\left(a^{2}–ab+b^{2}\right)[/latex]?

Example

Multiply [latex](a+b)(a^{2}–ab+b^{2})[/latex].

Did you see that? Four of the terms cancelled out, leaving us with a binomial [latex]a^{3}+b^{3}[/latex] which is a sum of cubes. To undo this process and get it back to the form we had originally, we need to factor [latex] a^3+b^3 [/latex].

Because of the above example, it may not be surprising that [latex]a^{3}+b^{3}[/latex] factors to be [latex](a+b)(a^{2}–ab+b^{2})[/latex] but how do we factor it?

You can use this pattern to factor sum of cubes: [latex]a^{3}+b^{3}=(a+b)(a^{2}–ab+b^{2})[/latex].

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].

Examples

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].

Let’s look at some detailed examples to figure out how we can get sum or differences of cubes into factored form.

Example

Factor [latex]x^{3}+8y^{3}[/latex].

You should always look for a greatest common factor (GCF) before doing any other factoring.

Example

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].

Examples

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].

[latex]a^{3}+b^{3}[/latex]=[latex]\left(a+b\right)\left(a^{2}-ab+b^{2}\right)[/latex]

[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 to use SOAP

SOAP is a memory device used for factoring sum or differences of cubes, to help you remember which signs you should use on each term in factored form.

“S” stands for Same
“O” stands for Opposite
“AP” stands for Always Positive

[latex]a^{3}+b^{3}[/latex]=[latex]\left(a+b\right)\left(a^{2}-ab+b^{2}\right)[/latex]
[latex]a^{3}-b^{3}[/latex]= [latex]\left(a-b\right)\left(a^{2}+ab+b^{2}\right)[/latex]

Pay close attention to the signs when you are changing it to factored form.

SAME:
[latex]a^{3}\color{red}{+}\color{black}{b^{3}=(a}\color{red}{+}\color{black}{b)(a^{2}-ab+b^{2})}[/latex]
[latex]a^{3}\color{red}{-}\color{black}{b^{3}=(a}\color{red}{-}\color{black}{b)(a^{2}-ab+b^{2})}[/latex]

OPPOSITE:
[latex]a^{3}\color{red}{+}\color{black}{b^{3}=(a+b)(a^{2}}\color{red}{-}\color{black}{ab+b^{2})}[/latex]
[latex]a^{3}\color{red}{-}\color{black}{b^{3}=(a+b)(a^{2}}\color{red}{+}\color{black}{ab+b^{2})}[/latex]

ALWAYS POSITIVE: the third term of the trinomial will always be positive because you are squaring that term. A negative times a negative is always positive and a positive times a positive is always positive. [latex]a^{3}+b^{3}=(a+b)(a^{2}-ab\color{red}{+}\color{black}{b^{2})}[/latex] and [latex]a^{3}-b^{3}=(a-b)(a^{2}+ab\color{red}{+}\color{black}{b^{2})}[/latex]

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

Example

Factor [latex]8x^{3}–1,000[/latex] completely.

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].

Example

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.

Review of Factoring

In a previous section, you have learned several factoring techniques. Now it is time to put it all together.

We focus on two important questions you should ask yourself when encountering any factoring problem:

  1. Which factoring technique should I use for this problem?
  2. Can I factor the polynomial more?

Choosing the Best Factoring Technique

Here, we present a strategy you can apply to any factoring problem. Refer back to Section 3.5 if you need more practice with factoring.

Factoring strategy

1. If there is a GCF other than [latex]1[/latex], factor it out first. Don’t forget to factor out a [latex]-1[/latex] if the leading coefficient is negative.

2. Count the number of terms in the remaining polynomial and select an appropriate technique.

I.  4 Terms:  Factor by Grouping

II.  3 Terms: [latex] ax^2+bx+c [/latex]

A. If [latex]a=1[/latex], apply the “Product and Sum Method.” Ask yourself: What multiplies to be [latex] c [/latex] that adds to be [latex] b [/latex]?

B. If [latex]a\neq 1[/latex], apply the “AC-Method”

C. If it is a perfect square trinomial, use the appropriate formula:

[latex]a^2+2ab+b^2=(a+b)^2[/latex]

[latex]a^2-2ab+b^2=(a-b)^2[/latex]

III.  2 Terms:

A. If the binomial is a difference of squares, use the following formula: [latex]a^2-b^2=(a+b)(a-b)[/latex]

 Remember that a sum of squares is not factorable: [latex] a^2 + b^2 [/latex] is a prime polynomial

B. If the binomial is a sum or difference of cubes use:

[latex]a^{3}+b^{3}[/latex]=[latex]\left(a+b\right)\left(a^{2}-ab+b^{2}\right)[/latex]

[latex]a^{3}-b^{3}[/latex]= [latex]\left(a-b\right)\left(a^{2}+ab+b^{2}\right)[/latex]

 

3. If none of the above applies, it is possible that the polynomial is not factorable, or “prime.”

Let us try some examples.

Example 1

Factor [latex]-3x^2-3x+6[/latex].

Example 2

Factor [latex]-3x^2-7x+6[/latex].

The next example includes a perfect square trinomial.

Example 3

Factor [latex]12x^5+60x^4+75x^3[/latex]

In the next two examples, we review factoring binomials.

Example 4

Factor [latex]-2x^2+8[/latex]

 

Example 5

Factor [latex]50x^{2}y^{3}-8y[/latex]

Don’t forget that we cannot factor every polynomial.

Example 6

Factor [latex]5x^2+8x+4[/latex]

Factoring More

Sometimes after we factor, one or more of the resulting polynomials can be factored even further. You will see this more in future classes, but present one example here for you to think about.

think about it

Factor completely: [latex]x^4-16[/latex]