Use factor theorem to prove that $(x+a)$ is a factor of $(x^n+a^n)$ for any odd positive integer n .
To do : Use factor theorem to prove that$(x+a) $ is a factor of $(x^n+a^n)$ for any odd positive integer n
Solution:
According to factor theorem, if $f(x)$ is a polynomial of degree n ≥ 1 and 'a' is any real number, then, $(x-a)$ is a factor of $f(x), \ if \ f(a)=0$.
Let $p(x) = x^n + a^n$ , where n is odd positive integer.
Take $(x+a)= 0$
$=> x = -a$
Consider:
$p(-a) = (-a) ^n + (a) ^n$
=$ -a^n + a^n$
= 0
Since, n is odd, by Factor theorem,
$(x+a)$ is a factor of $p(x)$ when n is odd positive integer .
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