Let $f=f(x, y)$ be an irreducible polynomial of degree $n \geq 2$ (over an algebraically closed field of characteristic zero) and $V_{0}=\{f=0\} \subset \mathbb{A}^{2}$ the corresponding affine plane curve. Assume that $V_{0}$ is smooth (non-singular) and that the projectivization $V \subset \mathbb{P}^{2}$ of $V_{0}$ intersects the line at infinity $\mathbb{P}^{2}-\mathbb{A}^{2}$ in $n$ distinct points. Show that $V$ is smooth and determine the divisor of the rational differential $\omega=\frac{d x}{f_{y}^{\prime}}$ on $V$. Deduce a formula for the genus of $V$.