By experimenting I found what seems to be true for polynomials in $\mathbb Z[X]$:
$\displaystyle \sum_{i=0}^n a_iX^i$ is irreducible iff $\displaystyle\sum_{i=0}^n b_iX^i$ is irreducible, where $b_{2\cdot j}=a_{2\cdot j}$ and $b_{2j+1}=\pm a_{2j+1}$.
I don't remember if I have seen this along time ago or not. I want a link or something.