How do we prove that (x+a) is a factor of x^n+a^n for all odd positive integer n?

1 Answer
Apr 25, 2018

Please see below.

Explanation:

According to factor theorem, if x-k is a factor of a polynomial function f(x), then f(k)=0

Now to test for x+a being a factor of f(x)=x^n+a^n,

as x+a=x-(-a), we must check for f(-a)

Now f(x)=x^n+a^n

Hence, f(-a)=(-a)^n+a^n

Observe if n is even then f(-a)=(-a)^n+a^n=a^n+a^n=2a^n!=0 and hence x+a is not a factor of x^n+a^n if n is even

but if n is odd then f(-a)=(-a)^n+a^n=-a^n+a^n=0

and hence x+a is a factor of x^n+a^n if n is odd.