Alternative complex analysis proof of Fundamental Theorem of Algebra My classmate and I working on the following problem from Conway's Functions of a Complex Variable I (p. 100) and we're completely stumped as to how we can do the problem without being circular.  The problem reads:
"Use Theorem 7.2 to give another proof of the Fundamental Theorem of Algebra."
Theorem 7.2 reads (with definitions inserted in square brackets):
Let $G$ be a region [an open, connected subset of the complex plane] and let $f$ be an analytic function on $G$ with zeros $a_1,\dots,a_m$ (repeated according to multiplicity).  If $\gamma$ is a closed rectifiable curve in $G$ [closed path of bounded variation] which does not pass through any point $a_k$ and if $\gamma \approx 0$ in $G$ [the winding number $n(\gamma;w)$ is zero for all $w \in \mathbb{C} \backslash G$] then $$\dfrac{1}{2 \pi i} \int_{\gamma} \dfrac{f'(z)}{f(z)} \; dz = \sum_{k=1}^m n(\gamma;a_k).$$
The only thing we could come up with was to go about the proof by contradiction, but since the formula in the theorem depends on knowing the $a_k$ are zeros, this doesn't have much promise...  Thanks in advance for your help.

Let $\gamma_R\colon [0,2\pi] \to \mathbb{C};\, \gamma_R(t) = R\cdot e^{it}$. By elementary estimates, one sees that for a polynomial $p$ of degree $n > 0$, there is an $R_0\in (0,+\infty)$ such that $p$ has no zeros outside the open disk $D_{R_0}(0)$. Cauchy's integral theorem says that hence
$$I(R) = \frac{1}{2\pi i}\int_{\gamma_R} \frac{p'(z)}{p(z)}\,dz$$
is constant on $[R_0,+\infty)$. Use the standard estimate (ML estimate) to see that
$$\lim_{R\to\infty} \int_{\gamma_R} \frac{p'(z)}{p(z)} - \frac{n}{z}\,dz = 0.$$
