Can every polynomial be moved to a irreducible one?

Question

Given a polynomial $P(x)$ in $Q[x]$, can I always find a rational number, $c$, so that $P(x)+c$ is irreducible in $Q[x]$?

Answer 1

I just recalled a neat lemma which solves this: Like Travis said we first move to $Z[x]$ and use the following lemma:

If a polynomial $P=a_nx^n+..+a_0 $ over $Z$ satisfies $a_0$ is prime and $|a_0|>|a_n|+|a_{n-1}|..+|a_1| $ then it is irreducible. A proof can be found here: http://yufeizhao.com/olympiad/intpoly.pdf page 2