# Use factor theorem to prove that

Question:

Use factor theorem to prove that $(x+a)$ is a factor of $\left(x^{n}+a^{n}\right)$ for any odd positive integer.

Solution:

Let $f(x)=x^{n}+a^{n}$

Putting $x=-a$ in $f(x)$, we get

$f(-a)=(-a)^{n}+a^{n}$

If n is any odd positive integer, then

$f(-a)=(-a)^{n}+a^{n}=-a^{n}+a^{n}=0$

Therefore, by factor theorem, $(x+a)$ is a factor of $\left(x^{n}+a^{n}\right)$ for any odd positive integer.