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Let $q$ be a polynomial of degree $n$ with distinct zeroes $z_1,\ldots,z_n$. Let $p$ be a polynomial of degree $n-2$ or less. Show that:

$\displaystyle\sum_{K=1}^{n} \operatorname{Res}\left(\frac{p}{q};z_k\right)=0$

I'm trying to think of ways to use the residue theorem, but I'm not sure how to proceed...any input would be appreciated!!

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Hint integrate over a really large circle.

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  • $\begingroup$ I was going to ask a similar question. I still don't understand how to proceed. How do we proceed? $\endgroup$ – User69127 Mar 24 '14 at 7:44
  • $\begingroup$ @User69127 Use the "ML-inequality" to show that $$\lim_{R\to\infty} \int_{|z|=R} \frac{p(z)}{q(z)}\, dz = 0.$$ $\endgroup$ – mrf Mar 24 '14 at 9:13

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