This question is related to mathematical expansions

Question

Can I get a proof of following Statement For every natural number n, $ (x^n - a^n) $ is always divisible by $ (x-a) $.

Answer 1

$(x^n-a^n)=(x-a)(x^{n-1}+x^{n-2}a+...+xa^{n-2}+a^{n-1}).$

This is a case of the Polynomial remainder theorem, where $f(x)=x^n-a^n:$

$f(a)=0,$ so the remainder when $f(x)$ is divided by $(x-a)$ is $0$.