Is $\{x f(x)+3g(x) \;|\;f,g\in \mathbb{Q}[x]\}$ a (main) ideal?

Question

Is it possible to show whether or not $ \{xf(x)+3g(x)\;|\;f(x),g(x) \in \mathbb{Q}[x]\} $ is an ideal (or main ideal in $\mathbb{Q}[x]$)?

I know how to prove it for $\mathbb{Z}[x]$, but what with $\mathbb{Q}[x]$?

Answer 1

Note that $3$ is invertible in $\mathbb{Q}[X]$. There is thus a very simple description for your set, which will allow to answer the question directly.

Answer 2

The ring $\;\Bbb F[x]\;$ is a principal ideal ring (PID) iff $\;\Bbb F\;$ is a field, so any ideal you find in $\;\Bbb Q[x]\;$ is principal.

OTOH, $\;\Bbb Z[x]\;$ is obviously not principal (since $\;\Bbb Z\;$ is not a field), so I'm not sure what you meant by " knowing to prove it in $\;\Bbb Z[x]\;$" .