How to prove a polynomial function with complex root is coercive?

Question

Let $p(z)$ be a polynomial of degree $n$, and $z$ is any complex number which can be written as $z=x+iy$. Show that $|p(z)|$ is coercive.

A function is called coersive if $f(x) \rightarrow +\infty$ as $||x|| \rightarrow \infty$.

Answer 1

$$p(z)=\sum_{k=1}^na_kz^k\;,\;\;n\in\Bbb N\,,\,\,a_n\neq0\implies p(z)=z^n\sum_{k=1}^n a_kz^{k-n}\implies$$

$$|p(z)|=|z|^n\left|a_n+\frac{a_{n-1}}z+\ldots+\frac{a_0}{z^n}\right|\le|z|^n\left(|a_n|+\frac{|a_{n-1}|}{|z|}+\ldots+\frac{|a_0|}{|z|^n}\right)\xrightarrow[|z|\to\infty]{}\infty$$

$$\text{since }\;\;\frac{|a_{n-k}|}{|z|^k}\xrightarrow[|z|\to\infty]{}0\;,\;\;\forall\,k\ge1$$