Let $\displaystyle (b_0,b_1,b_2,b_3)$ be a permutation of the set $\displaystyle \{54,72,36,108\}$. Prove that $\displaystyle x^5+b_3x^3+b_2x^2+b_1x+b_0$ is irreducible in $\displaystyle \mathbb Z[x]$.
This was given me by my teacher in a problem sheet, I haven't done anything. I hope that there is a solution which does not consider $24$ permutations separately. A slick solution is welcomed.