# Let Q(z)=(z−α_1)⋯(z−α_n) be a polynomial of degree >1 with distinct roots outside the real line. sum_(j=1)^n /(1)(Q'(alpha_j))=0. Do we have a proof relying on rudimentary techniques?

Let $Q\left(z\right)=\left(z-{\alpha }_{1}\right)\cdots \left(z-{\alpha }_{n}\right)$ be a polynomial of degree $>1$ with distinct roots outside the real line.
We have
$\sum _{j=1}^{n}\frac{1}{{Q}^{\prime }\left({\alpha }_{j}\right)}=0.$
Do we have a proof relying on rudimentary techniques?
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Micah Hobbs
Let $n\ge 1$ and ${\alpha }_{1},\cdots ,{\alpha }_{n}\in \mathbb{C}$ be distinct. If we put $Q\left(z\right)=\left(z-{\alpha }_{1}\right)\cdots \left(z-{\alpha }_{n}\right)$, then from the partial fraction decomposition, it follows that
$\frac{1}{Q\left(z\right)}=\sum _{j=1}^{n}\frac{1}{{Q}^{\prime }\left({\alpha }_{j}\right)\left(z-{\alpha }_{j}\right)}.$
From this, we have