$x-y$ divides $x^n - y^n$ -- prove by mathematical induction

Question

I'm doing some prove by mathematical induction practice problems. I got to this last problem but I don't know how to approach it.

This is the question

For all positive integers $n$ and any distinct real numbers $x$ and $y$, $x - y$ divides $x^n - y^n$.

Answer 1

Using strong induction:

Base cases $n=1$ and $n=2$ hold trivially $(x-y)\mid (x-y)$ and $(x-y)\mid (x^2-y^2)$ since $x^2-y^2=(x+y)(x-y)$

Assume for our induction hypothesis that the claim $(x-y)\mid (x^k-y^k)$ holds for all $k$ up to $n$ for some $n$.

Consider $n+1$

$x^{n+1}-y^{n+1}=(x+y-y)x^n-(y+x-x)y^n$

$=(x+y)(x^n-y^n)-yx^n+xy^n=(x+y)(x^n-y^n)-xy(x^{n-1}-y^{n-1})$

Each term of which is divisible by $x-y$ by our induction hypothesis, thus the hypothesis is validated and the claim is true for all positive integers $n$.

Answer 2

Compute $(x-y) \sum_{k=0}^n x^k y^{n-k}$.

Answer 3

Hint: $$x^{n+1}-y^{n+1}=x^{n+1}-x^ny+x^ny-y^{n+1}=x^n(x-y)+y(x^n-y^n).$$

Remark: You don't really want to say any distinct real numbers, for any non-zero real divides any real number, so there is nothing to prove. Integers is fine.