# Thread: Proving a factor of a polynomial

1. ## Proving a factor of a polynomial

Hi,

I hope someone can help provide elaboration on the homework question that I'm faced with, and why the solution is what it is.

The homework question is as follows:
Determine a general rules to help decide whether x - a and x + a are factors of x^n - a^n and x^n + a^n.

The homework solution is as follows:
- For x^n - a^n, if n is even, they're both factors.
- If n is odd, only x - a is a factor. For x^n + a^n, if n is even, neither is a factor.
- If n is odd, only x + a is a factor.

Sincerely,
Olivia

2. ## Re: Proving a factor of a polynomial

the factor theorem states that

$(x-a) \text{ divides } p(x) \Rightarrow p(a)=0$

consider $x^n-a^n=0$

If $n$ is even, then both $x=\pm a$ satisfy this equation and thus both $(x-a)$ and $(x+a)$ are factors of $p(x)$

If $n$ is odd, then only $x=a$ satisfies this equation and thus only $(x-a)$ is a factor of $p(x)$

Apply this same reasoning to $x^n + a^n$ to obtain the other two results.

I leave that to you.

3. ## Re: Proving a factor of a polynomial

Thank you That makes total sense now.