I was attempting to simplify a standard proof of the fundamental theorem of algebra for undergraduates who know about Taylor series and some complex numbers. Normally, the proof would look like:
Assume $f(z)$ is a polynomial that is not equal to zero anywhere. Since $f(z)$ has no zeros, $\frac{1}{f(z)}$ is an entire function and thus has a Taylor series valid for all $z\in\mathbb{Z}$. From here ...
However, I would like to prove that $\frac{1}{f(z)}$ has a Taylor series everywhere (with a non-zero radius of convergence) without appealing to the fact that the inverse of an entire function with no zeros is still an entire function. That is, I would prefer not to name/talk about entire functions all together. Is there a way to do this using only basic calculus (definitely using the fact that $f(z)\neq 0$) and perhaps some beginner facts about complex numbers (they can be added, subtracted, multiplied, divided, and magnitude perhaps) in a clever manner?
EDIT: If anybody has a solution but they are not sure if it is "simple" enough then feel free to post it as an answer. There are no hard and fast rules to what is simple so anything might work.