Difference of Even PowersReference > Mathematics > Algebra > Factoring Higher Degree Polynomials
If you've followed through from the beginning of this unit, you've already learned how to factor a difference of squares, and you've already learned how to factor a difference of cubes or a sum of cubes, and you've learned how to factor a sum or difference of any odd power (assuming both terms have the same odd power, of course!) In learning all this, you've already learned how to factor the difference of any even power. It's true! You may not have realized it, but you have all the tools you need. Let me show you with an example:
Factor the following: x6 - y6
Since the exponents are even, it's automatically a difference of squares, so we use the difference of squares rule: x6 - y6 = (x3 - y3)(x3 + y3)
Now, using difference of cubes rule and sum of cubes rule, this factors into (x - y)(x2 + xy + y2)(x + y)(x2 - xy + y2)
There! Wasn't that easy?
Here's one that's a little more complex:
Factor the following: 2048x11 - 2x
Since we can factor 2x out of both terms, we have 2048x11 - 2x = 2x(1024x10 - 1).
The second factor is a difference of squares so we can rewrite this as 2x(32x5 - 1)(32x5 + 1).
(32x5 - 1) is a difference of fifth powers, and (32x5 + 1) is a sum of fifth powers, so the entire expression factors like this:
2x(2x - 1)(16x4 + 8x3 + 4x2 + 2x + 1)(2x + 1)(16x4 - 8x3 + 4x2 - 2x + 1)