meromorphic at $\infty$ is a polynomial?

Question

I do not understand the statement why $p$ will be polynomial in the following statement:

"The function $p:\hat{\mathbb C}\rightarrow\hat{\mathbb C}$ defined by $p(z)=f(z)q(z)$ has the removable singularities at the poles of $f$ in $\mathbb C$, so it is entire, this it has power series representation on all $\mathbb C$, also $p$ is meromorphic at $\infty$ as both $f$ and $q$ are.this forces $p$ to be a polynomial.Since, $f=\frac pq$" $$ q(z)=\prod_{j=1}^{n}(z-z_j)^{e_j}$$

Answer 1

What does it mean to have a pole at $\infty$? This means that the function has a pole when written in terms of the coordinate $w=\frac1z$ .