Irreducible polynomials have no zeroes with multiplicity $\geq 1$

Question

Let $0\neq f\in k[X_0,\ldots,X_n]$ be an irreducible polynomial with coefficients in the algebraically closed field $k$. I'm trying to check that there exist $n$ points $x_1,\ldots,x_n\in k$ such that $0\neq f(X_0,x_1,\ldots,x_n)$ has no zeroes with multiplicity greater thatn $1$. Any hint?