Whether $f(x)$ is reducible in $ \mathbb Z[x] $? Suppose that $f(x) \in \mathbb Z[x] $ has an integer root. Does it mean $f(x)$ is reducible in $\mathbb Z[x]$?

If the degree of the polynomial is 1, then it is irreductible by definition. But if the degree is greater than one and it has an integer root $r$, then the polynomial is reductible, because it can be written in the form $f(x)=(x-r)g(x)+r(x)$, where $r(x)$ is a constant (by remainder division theorem). Plugging $x=r$ yields that $r(x)=0$ and therefore $f(x)=(x-r)g(x)$.
