Meromorphic function at infinity

Question

I'm studing about meremorphic functions in complex analysis and I'm trying to understand the proofe of being a mereomorphic function on $\mathbb{P}^1(\mathbb{C})$ means the function is rational. I also read what went on a question on meromorphic function at $\infty$ but what I don't understand is what happens if infinity is one of the poles. In that case, we can't take $q(z)=\prod_{j=1}^n(z-z_j), z_j$ are the poles, as they did in the proofe in the link above, since $z_j=\infty$. How is that setteled?