Prove the Fundamental Theorem of Algebra using Liouville

Let \[ f(z)=a_0+a_1z+\cdots+a_nz^n, \qquad n\ge1,\quad a_n\ne0. \] Prove that there exists \(z\in\mathbb C\) such that \(f(z)=0\).

Hint: use Liouville’s theorem.