Irreducible polynomial for infinitely many values

Question

I want to prove that there are infinitely many values of k such that the polynomial $x^{9}+12x^{5}-21x+k$ is irreducible. I sense that I have to use Eisenstein and the number 3 but I don't see exactly how. Any help would be appreciated.

Answer 1

General idea: What does Eisenstein with the prime $3$ say if $k=3$? What about $k=6$? What about $k=9$? Can you now find infinitely many $k$ that work?