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.
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