Greene and Krantz pose the following problem in Function Theory of One Complex Variable, Ch. 5 problem 3:
Give another proof of the fundamental theorem of algebra as follows: Let $P(z)$ be a non-constant polynomial. Fix $Q\in \mathbb{C}$. Consider \begin{equation} \frac{1}{2\pi i} \oint_{\partial D(Q,R)} \frac{P'(z)}{P(z)}\,dz. \end{equation} Argue that as $R\to +\infty$, this expression tends to a nonzero constant.
I was thinking along these lines: Since we do not know $P(z)$ factors completely, let us write $ P(z) = \prod_j (z - \alpha_j) \, g(z),$ where $g(z)$ is an irreducible polynomial. Now $ \frac{P'(z)}{P(z)} = \sum_k \frac{1}{z-\alpha_k} + \frac{g'(z)}{g(z)}.$ Each of the terms $1/(z-\alpha_k)$ adds $1$ to the integral expression. As $R \to \infty$, all the $\alpha_k$ are eventually inside $D(Q,R)$, whereas the term $g'(z)/g(z)$ approaches zero, since the denominator has a higher degree. Is the reasoning correct ? Can someone offer a simpler argument ?