Is there a "nice" formula for the product $\prod_{n\neq i}(x-z_n)$ which contains all roots of unity except exactly one?

Question

Let $Q_p(x)=x^p-1$, $p$ is an odd prime.
I was wondering if there is some nice formula for the product $\prod_{n\neq i}(x-z_n)$ which contains all except one (let's say $z_i$) $p$-th roots of unity.
For example if $p=5$ and $z_1=e^{\frac{2\pi i}{5}}$, is it possible to express $\frac{Q_5(x)}{x-z_1}=(x-1)(x-z_2)(x-z_3)(x-z_4)$ in a different way? I tried to express all $5$-th primitive roots $z_2, z_3, z_4$ as powers of $z_1$ but without much success.

Answer 1

It's $$\frac{x^p-1}{x-\zeta}$$ where $\zeta=z_n$ is a $p$-th root of $1$. Then $$\frac{x^p-1}{x-\zeta}=\frac{x^p-\zeta^p}{x-\zeta}=x^{p-1}+\zeta x^{p-2}+ \zeta^2x^{p-3}+\cdots+\zeta^{p-2}x+\zeta^{p-1}.$$