Compute $$ \lim_{R \to \infty} \frac{1}{2\pi i} \int_{|z|=R} \frac{(2z^2+z-1)P'(z)}{P(z)+3}dz $$ where $P(z)=z^{10}+2z^9+z^5+1$.
It seems like I may use residue but it contains too high order polynomials to do that efficiently? How should I do it instead?
Let $Q(z)=P(z)+3=z^{10}+2z^9+z^5+4$. We have: $$\lim_{R\to +\infty}\frac{1}{2\pi i}\oint_{|z|=R}(2z^2+z-1)\frac{Q'(z)}{Q(z)}\,dz=\sum_{\xi\in Z}(2\xi^2+\xi-1) $$ where $Z$ is the set of zeroes of $Q(z)$. Since, by Viète's theorem, $$\sum_{\xi\in Z}\xi = -2,\qquad \sum_{\xi\in Z}\xi^2 = (-2)^2-2\cdot 0 =4,$$ we simply have: $$\lim_{R\to +\infty}\frac{1}{2\pi i}\oint_{|z|=R}(2z^2+z-1)\frac{Q'(z)}{Q(z)}\,dz = 8-2-10 = \color{red}{-4}.$$