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I need help in proving the following.

Given $Q$ a polynomial evaluated over the complex numbers such that $\alpha_1, ..., \alpha_n$ are roots of $Q$ and $P$ a polynomial such that $\deg P<n$ prove that $$ \frac{P(z)}{Q(z)}=\sum_{k=1}^{n}\frac{P(\alpha_k)}{Q'(z)(z-\alpha_k)} $$ At first I tried kind of 'brute forcing' from one side and then the other and I think I got farther when beginning with the right side but still I didn't manage to get the identity. Any help is appreciated.

D. Brito
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    As in @dxiv's comment... note you have a typo in the denominator - $Q'(\alpha_k)$ rather than $Q'(z)$. Also the $\alpha_j$ should be distinct. Finally, for a purely algebraic proof, it follows from Lagrange interpolation - see https://en.wikipedia.org/wiki/Lagrange_polynomial, for instance (one gets an expression for $P(z)$; dividing it by $Q(z)$ gives you the result you want). – peter a g Sep 26 '18 at 01:52

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