Solutions of $1+x+x^2+...+x^n$

Question

I was thinking about the roots of unity, the solutions of the polynomial $P(x) = x^n - 1$ which are quite easy to find, they are of the form $ \cos \frac{2k\pi}{n} + i\sin \frac{2k\pi}{n} $. I was wondering if there are any other polynomials whose solutions we know. Particularly, is there any way to find the solutions of $ P(x) = 1 + x^2 + x^3 + \dots + x^n $?

Answer 1

Observe that

$x^{n + 1} - 1 = (x - 1)(x^n + x^{n - 1} + \ldots + 1); \tag 1$

therefore any root of

$x^{n + 1} - 1 = 0 \tag 2$

other than $x = 1$ is a root of

$x^n + x^{n - 1} + \ldots + 1 = 0; \tag 3$

thus the $n$ roots of (3) are

$x = e^{2k\pi i / (n + 1)}, \; 1 \le k \le n. \tag 4$