How do i check if a polynomial is irreducible?

Question

I have this excercise. let ${ f=2x \in R[x] }$. Determine the irreducibility of the polynomial ${f}$ if ${ R= \mathbb{Z} }$ or ${ R= \mathbb{Q} }$. Im not really sure how to do it and how it would change in the two domains that theyre giving me .

Answer 1

In $\mathbb{Z}[x]$, $f$ is reducible, as $f(x)=g(x)h(x)$ for $g(x)=2$, $h(x)=x$ and both these polynomials are non-zero and non-units in $\mathbb{Z}[x]$.

In $\mathbb{Q}[x]$, $f$ is irreducible, as if we write $f(x)=g(x)h(x)$, one of the two polynomials must be a non-zero constant, which is a unit in $\mathbb{Q}[x]$, as $\mathbb{Q}$ is a field.