## Factoring a Difference of Cubes or a Sum of Cubes

Many students find factoring a difference of cubes (or a sum of cubes) to be difficult. When I ask my students what they find so difficult about the process, they say they don’t understand the formula and how to use it. I have come up with a way to help my students remember the formula. Here’s how I teach the topic.

### Identification

Before you can factor a difference of cubes, you must be able to identify the expression as a difference of cubes. We are looking for a binomial (2 terms) that is a difference. Each exponent must be a multiple of 3, so the term can be expressed a cube. Each coefficient or constant must be a perfect cube: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …

### Formula

The formula for factoring $A^3 - B^3$ is $(A - B) (A^2 + AB + B^2)$. Here’s how I teach it.

I have my students begin by rewriting the binomial as a difference of cubes.

$x^3 - 343 = (x)^3 - (7)^3$

To determine the binomial factor, I tell my students to “drop the cubes”. So, the factoring begins with $(x)^3 - (7)^3 = (x-7)(......)$.

Now, on to the trinomial factor. Instead of worrying about the formula, I tell my students to write the first term (x) three times and the second term (7) three times:

xxx777

They remember to write them 3 times because they associate the number 3 with cubes. Now I have them group them into pairs:

(xx) (x7) (77)

$(xx) = x^2, (x7) = 7x, (77) = 49$

Now I have the three terms for my trinomial: $x^2$, $7x$, and $49$. All I need to figure out are the signs of the second and third terms. The sign of the second term is always the opposite of the sign in the original problem, and the sign of the third term is always positive.

$x^3 - 343 = (x)^3 - (7)^3 = (x-7) (x^2 + 7x +49)$

The same process works for sum of cubes as well. We find the terms the same exact way, and the rule for the signs is the same as well.

If you have any questions about factoring, or if you have any techniques that you use, I’d like to encourage you to share by leaving a comment, or reaching me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular math topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

• 1. georgewoodbury  |  May 13, 2010 at 4:03 pm

My friend Fred shares this suggestion for remembering the signs: The sign in the binomial factor is the Same (S) as the sign in the original problem. The next sign is the Opposite (O), and the last sign is Always Positive (AP). Factoring polynomials is a “dirty” business – use SOAP.

• 2. Mike Mcilveen  |  May 13, 2010 at 4:06 pm

That was very accessible! Anothef fav is apply factor theorem to f(x)=x^3-1, with x=1; use polynomial division to divide x^3-1 by x-1. Sounds like a lot but really gets to some interesting math that extends to diff of 5th powers and so on.

• 3. georgewoodbury  |  May 13, 2010 at 4:14 pm

I like that too Mike. In precalculus I do the same for x^3 – 1, x^4 – 1, x^5 – 1, … Then I have my students try to come up with formulas for factoring x^k – a^k. It’s a great exercise.

• 4. David Radcliffe  |  May 13, 2010 at 6:22 pm

I like this technique a lot, and I think that my students would find it helpful for memorizing the factorizations for sums and differences and cubes.

This is slightly off-topic, but related to the last sentence of your post. I gave my students an extra credit assignment in which I asked them to use WolframAlpha to factor x^n – 1 for different values of n, and comment on any patterns that they observed. I suggested that they consider what happens when n is even, odd, prime, or a power of 2. There were a few interesting observations, but not many. Next time I will add some more structure to the assignment.

• 5. georgewoodbury  |  May 13, 2010 at 6:50 pm

Love the W|A idea! It’s a great way to develop patterns and increase intuition. Discovery learning at its finest.

• 6. Tony  |  January 22, 2014 at 8:30 pm

I’m a student and this was very helpful. Thank you Professor Woodbury!