Factor x^5+y^5

Question

How do I get an answer of (x+y)(x^4 - x^3y + x^2y^2 - xy^3 + y^4)?

 

Answer 1

Edward, this is something you will almost have to memorize.  Let's look at how it works.

 

We can either have the difference or the sum of a and b, each to the same odd power, ie a³  - b³, or a⁹ + b⁹.

 

If you have a difference, then your first factor will be (a-b).  If you have a sum, the first factor will be (a+b).

 

The second factor works like this:  You start with a to the power of your power-1, so if you had a⁷, you'd start with a⁶  and work down to a power of 0 (a⁰ = 1, right?).

 

The b's go the opposite way.  You start with b⁰, which also equals 1, and go up to b to the power of your power-1.

 

If you have a difference, then everything in the second part will be positive. If you have a sum, then we start with positive and alternate signs. The reason for this is that all but one of our pairs must cancel out.

 

Let's look at your example.  x⁵ + y⁵ = (x+y)(x⁴y⁰ - x³y¹ + x²y² - x¹y³ + x⁰y⁴).

 

Since anything to the 0 power equals 1 and anything to the first power is itself, then we have

 

(x+y)(x⁴ - x³y + x²y² - xy³ + y⁴).

 

If we multiply that out, we get x⁵ + x⁴y - x⁴y - x³y² + x³y² + x²y³ - x²y³ - xy⁴ + xy⁴ + y⁵.

 

Everything in the middle cancels out and we end up with x⁵ + y⁵.

 

For fun, let's try x¹¹ - y¹¹.

 

Then, we'd have (x-y)(x¹⁰ + x⁹y + x⁸y² + x⁷y³ +x⁶y⁴ + x⁵y⁵ + x⁴y⁶ + x³y⁷ + x²y⁸ + xy⁹ + y¹⁰)

 

I hope this makes sense!