Calculate $(i+u_0)(i+u_1)\cdots(i+u_{n-1})$ if $u_0,u_1,...u_{n-1}$ are $n-$th roots of unity [closed]
Let $u_0,u_1,...u_{n-1}$ be all complex $n-$th roots of unity.
Calculate the product: $$(i+u_0)(i+u_1)\cdots(i+u_{n-1})$$ I've tried various tricks, like $i = e^{i\frac{1}{2}\pi}$, $u_k=e^{ik\frac{2}{n}\pi}$. Also tried to evaluate the product as a polynomial, but I'm not sure if it yields anything.
Note that, if $u_0,\ldots,u_{n-1}$ are roots of unity, then $$ x^n-1=\prod_{j=0}^{n-1}(x-u_j) $$ hence for $x=-i$, we obtain $$ (-i)^n-1=\prod_{j=0}^{n-1}(-i-u_j)=(-1^n)\prod_{j=0}^{n-1}(i+u_j) $$ and thus $$ \prod_{j=0}^{n-1}(i+u_j)=(-1)^n\big((-i)^n-1\big)=i^n-(-1)^n. $$