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

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?

• There is just one polynomial in your post. What do you want it to not have common factors with? – Lukas Kofler May 13 '18 at 23:17
• By "common factors" do you mean zeroes of multiplicity greater than one? If not that, what? Cheers! – Robert Lewis May 13 '18 at 23:23
• Yes @RobertLewis. I mean zeroes of multiplicity$\geq 1$. – Vincenzo Zaccaro May 13 '18 at 23:38
• We have complitely right! The question was not well written! – Vincenzo Zaccaro May 13 '18 at 23:40