Question

Use factor theorem to verify that x+a is a factor of $x^n+a^n$ for nay odd positive integer.

Answer 1

Let $f\left(x\right)=x^n+a^n$.

In order to prove that $x+a$ is a factor of $f\left(x\right)$ for any odd positive integer n, it is sufficient to show that $f(-a)=0$.

$f(-a)=(-a{)}^n+a^n=(-1{)}^n{a}^n+a^n$

$f(-a)=(-1+1)a^n$ [ n is odd positive integer ]

$f(-a)=0\times a^n=0$

Hence, $x+a$ is a factor of $x^n+a^n$, when n is an odd positive integer.