Sum or Difference of Odd Powers

Reference > Mathematics > Algebra > Factoring Higher Degree Polynomials

Having learned about both sum of cubes and difference of cubes in this unit, we're now going to take those two rules and extend them to any odd power (for example, a⁵ - b⁵, or a⁷ + b⁷). These may look intimidating, but once you've learned sum of cubes and difference of cubes, these are actually fairly straightforward. Let's start by looking at the example above: a⁵ - b⁵.

Difference of Odd Powers

a⁵ - b⁵ = (a - b)(a⁴ + a³b + a²b² + ab³ + b⁴)

This starts out just like a³ - b³, with the binomial (a - b). What follows is an extended form of the difference of cubes trinomial; in this polynomial, we started with a⁴ and worked our way down by degrees, meanwhile working up to b⁴ by degrees. The thing that makes this simple to keep track of is that every term in the second factor has degree 4. So if we add our exponents for a and b in each term, we should get four.

Example One
Factor the following: 128x⁷ - 2187

Solution
In this problem, we note that both terms are perfect seventh powers, so we write out the following:

a⁷ - b⁷ = (a - b)(a⁶ + a⁵b + a⁴b² + a³b³ + a²b⁴ + ab⁵ + b⁶)

Note that each term in the second factor has degree 6 (one less than 7).

We need to find a and b now. Since a⁷ = 128x⁷, and b⁷ = 2187, a = 2x, and b = 3. The factorization is:

(2x - 3)((2x)⁶ + (2x)⁵(3) + (2x)⁴(3)² + (2x)³(3)³ + (2x)²(3)⁴ + 2x(3)⁵ + 3⁶)
(2x - 3)(64x⁶ + 96x⁵ + 144x⁴ + 216x³ + 324x² + 486x + 729)

If I've done this right, the ratio between each coefficient with the successive one will be the same, and I can quickly verify this: If I start with 64, and keep multiplying it by

, I will get all of my coefficients in order. In addition, my exponents for x are decreasing from left to right.

Sum of Odd Powers

You might be able to guess what the sum of odd powers factorization will look like. Here I'll show you the one for a⁷ + b⁷.

a⁷ + b⁷ = (a + b)(a⁶ - a⁵b + a⁴b² - a³b³ + a²b⁴ - ab⁵ + b⁶)

As you can see, the binomial has a plus sign (just like in the sum of cubes rule - which completely makes sense, since cubing is raising to an odd power!). The second factor is just like the second factor for Difference of Odd Powers, except that every other term is negative.

Example Two
Factor the following: 64x¹⁰ + 2y⁵

Solution
First we note that 2 can be factored out of both terms, giving the following:

64x¹⁰ + 2y⁵ = 2(32x¹⁰ + y⁵)

Now we observe that both factors in the binomial are 5th powers. Thus, our factoring rule will be:

a⁵ + b⁵ = (a + b)(a⁴ - a³b + a²b² - ab³ + b⁴)

Since a⁵ = 32x¹⁰ and b⁵ = y⁵, a = 2x², and b = y. Now we plug these into our rule:

64x¹⁰ + 2y⁵ = 2(2x² + b)((2x²)⁴ - (2x²)³(y) + (2x²)²(y)² - (2x²)(y)³ + y⁴)

64x¹⁰ + 2y⁵ = 2(2x² + b)(16x⁸ - 8x⁶y + 4x⁴y² - 2x²y³ + y⁴)

Note that, as with the previous example, our exponents make a nice pattern, as do our coefficients. If they didn't we ought to be concerned that we've done something wrong!