Question on proof concerning ideals containing monomials and their varieties

Question

An ideal I in C[x] contains a monomial if and only if each point in its variety V(I) has at least one zero coordinate. I'm trying to prove the above statement. The "only if" direction is easy for me, but I'm pretty stuck on the "if" direction. I'm also quite new to algebraic geometry. Does anyone have any tips or tricks that could be used to make progress on this question? I have tried proof by contradiction but am not making progress. My general approach has been to introduce a Grobner basis and then show that one of the basis elements must be a monomial.

Answer 1

Hint: The condition that each point in $V(I)$ has at least one zero coordinate is equivalent to $V(I) \subseteq V(x_1 x_2 \cdots x_n)$.