Factoring
Sum or Difference of Cubes
On this page you will learn how to factor the sum or difference of cubes. This is an important skill in Algebra, and can play a huge role in finding the real zero (rational roots, real x – intercepts) of a polynomial function.
The process itself is pretty simple, once you determine what the values of a and b are respectively. To learn about what that means, please read through the notes below. Then, watch the video. Try the practice problems and even the quiz that is embedded below.
A good way to think of factoring is as the opposite of distributing. For example, if you had (x + 3)(x – 3), and you distributed, it would look like this:
(x + 3)(x – 3)
x(x – 3) + 3(x – 3)
x2 – 3x + 3x – 9
x2 – 9
So, if you were asked to factor x2 – 9, you’re being asked, what polynomials multiply together to make x2 – 9? The answer is of course (x + 3)(x – 3).
This example was pick with intent because it is an example of a difference of squares. The first and last terms are perfect squares, and they’re being subtracted. There’s a pattern to factoring these that is pretty straight forward. Let’s formalize that pattern, as it will tie something familiar with what’s new here.
In our first example, x2 – 9, a = 1x, and b = 3. Let’s see what that means.
For this difference of squares, 25x4 – 49y2, a = 5x2, and b = 7y. Factoring the difference of squares yields (5x2 – 7y)(5x2 + 7y). Let’s see that in a column.
In this section we will learn to factor the sum and difference of cubes.
Sum of Cubes
For our application here, a binomial can be the sum of cubes if:
- Each term is a cubed number
- Each term is positive
- The operation in the binomial is addition
In other words, the sum of cubes is a polynomial where a and b are positive, in the form:
Before we learn the pattern for factoring, let’s make sure we can identify the parts. If the parts are easily identified, the factoring is nothing more than plugging values into a formula. Ready?
In example 1, a = x and b = 1. In the second example, a = 2x2 and b = 3.
Here’s how the formula works:
Let’s carry out the multiplication on the right side of the equation above to verify that this is the correct factorization.
Let’s try an example now.
Difference of Cubes
The only thing different between the sum of cubes and the difference of cubes is the operation. Sum of cubes adds two cubed numbers. Difference of cubes subtracts two cubed numbers. Here are a few examples.
The formula for the difference of cubes is as follows.
To factor a difference of cubes:
- Identify a and b
- Plug a and b into the formula
Let’s try the trickiest problem above: 64 – 27x6. Here a = 4 because 43 = 64, and b = 3x2, because (3x2)3 = 27x6. Notice that a and b are still considered positive!
Tip: Notice how the relationship between the addition and subtraction signs in the sum and difference of cubes factoring formulas.
