I want to prove that the following polynomial is irreducible: $$x^3 - x^2 - x + 3$$ My question gives the hint to apply the substitution $x \mapsto x+1$ but I've tried this and when multiplied out I'm getting $x^3 + x^2 +2$. I tried this mod 2 but came out with a reducible polynomial ($x^3 + x^2$ which can be written $(x^2)(x+1)$).
Can anyone tell me what I'm missing? I know that irreducible mod prime number implies irreducible in $\mathbb Z$, but am I wrong in thinking it works the other way i.e. reducible mod prime means reducible in $\mathbb Z$?