Factorise $x^n+x^{n-1}+...+x+1$?

Question

Is there a way to factorise $x^n+x^{n-1}+. . .+x+1$? I've tried to take the $1$ out, but now I do not know where to go from here because it doesn't seem to work.

Answer 1

Hints:

The roots of this polynomial are the complex $n+1\mkern-1mu$-th roots of unity different from $1$. So, if you set $\zeta=\mathrm e^{\tfrac{2i\pi}{n+1}}$, it factors over $\mathbf C$ as $$(x-\zeta)(x-\zeta^2)\dots(x-\zeta^k)\dotsm(x-\zeta^n),$$ so all you have to obtain a factorisation over $\mathbf R$ as a product of irreduciblle quadratic polynomials is to group the factors corresponding to conjugate powers of $\zeta$.

You'll have to consider two cases: $n$ odd and $n$ even.