Prove that there are infinitely many values of $α$ for which $f(x)= x^{7} +15x^{2} − 30x + α$ is irreducible in Q[x]

Question

Prove that there are infinitely many values of α for which $ x^{7} +15x^{2} − 30x + α $ is irreducible in $Q[x]$. I know that an irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. I need a general way to show irreducibility. Thanks.

Answer 1

You can show with the Eisenstein-criterion that every $\alpha$ divisible by $5$ , but not divisible by $25$, does the job.