We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.



> 
> Is it true that for each irreducible $g(X) \in \mathbb{Q}[X]$ there exists some
>  nonzero $h(X) \in \mathbb{Q}[X]$ such that $g(X)h(X)$ is a trinomial?
> 
> 
> 


If no, then what are some necessary/sufficient conditions on $g$ for the existence of such $h$?



Is every polynomial a factor of a trinomial?
To summarize some of the discussion in the comments: a trinomial can have at most four distinct real roots. A random polynomial of degree $d$ (random = all coefficients are iid, there are other models) has $\Omega(\log d)$ real roots. Which means that the probability that a random polynomial divides a trinomial goes to zero (exponentially fast with degree, though this is a little harder to prove).


