Prove the following:
If $k$ is an algebraically closed field and $f(x_1, \ldots, x_n) \in k[x_1,\ldots, x_n]$ is non-zero, then there exists $(a_1, \ldots, a_n)\in k^n$ s.t $f(a_1, \ldots, a_n) = 0$. I am supposed to prove this without the use of any large theorem and just from elementary principles.
I thought I had a proof for it, but realized that it did not cover all cases. I then tried an inductive argument, but it wasn't working as well as I had hoped.