if $1, z_1, ... , z_{n-1}$ are the solutions to $z^n = 1$, prove that $(z - z_1)(z - z_2)\dotsb(z - z_{n-1}) = 1 + z + z^2 + \dotsb + z^{n-1}$

Question

I've been trying this problem but i cannot do it only with complex variable (that's the idea), it always shows up cycling polynomials and I cannot use that.

Edit: I'v been struggiling with the fact that $\frac{z^n - 1}{z-1} = \sum_{k=0}^{n-1}z^k$, and i cannot prove that

Answer 1

Hint The product $$\prod_{i=1}^n (x-r_i)$$ over the roots of a monic polynomial $p(x)$ equals $p(x)$.