I am trying to prove that the winding number $n(p\circ\gamma_r,0)$ is $k$, when $p(z)=a\prod_{j=1}^k(z-z_j)$ is a complex polynomial of degree $k$ and $\gamma_r:[0,2\pi] \rightarrow \mathbb{C}, \gamma_r(t)=re^{it} $. For $r$ we assume $r > max_{j=1, \dots, k} |z_j|$.
So far I have tried to compute $\frac{1}{2\pi i}\int_{p\circ\gamma_r}\frac{dw}{w} $ and arrived after some calculations at $$\frac{1}{2\pi i}\int_{p\circ\gamma_r}\frac{dw}{w}=\frac{1}{2\pi i}\int_{0}^{2\pi}\frac{f'_1}{f_1}(t)+\dotsi+\frac{f'_k}{f_k}(t)dt$$ where $f_j= re^{it}-z_j$ and $f'_j=ire^{it}$. However, I could not produce any idea how to evaluate the antiderivatives of the quotients $\frac{f'_j}{f_j}(t)$. Or am I on the wrong path anyway? Any hint would be highly appreciated.
In the comments I learned that the residue theorem and some properties of homotopy of closed curves are very helpful here, but unfortunately, neither of these topics has been subject of our lectures yet. So I would be very greatful if someone could give me a hint on how to proceed without these theorems.