Question

If n is odd, then prove that x+1 is a factor of x^n+1

Answer 1

If n is odd then x^{n} +1 can be factorised as
(x+1) (1-x+x^{2}-x^{3} +...... +x^{n-1}).
Thus x+1 is a factor.

We can prove it by factor theorem also as follows:
Take p(x)=x^{n} +1.
When n is odd, we get
p(-1)= -1+1=0.
Hence x+1 is a factor of p(x).