Monic polynomal $f \in \mathbb{Z}[X]$ such that $f(2)=13$ have at most three distinct roots in $\mathbb{Q}$?

Question

how can I show that given a monic polynomial $f \in \mathbb{Z}[X]$ such that $f(2)=13$ have at most three distinct roots in $\mathbb{Q}$?

I know that these roots has to be in $\mathbb{Z}$, but I don't know how to continue or how to use this.

Answer 1

If $f(X)$ has four distinct roots $a,b,c,d$, then $(X-a)(X-b)(X-c)(X-d)$ divides $f(X)$, so $(2-a)(2-b)(2-c)(2-d)$ must divide $f(2)=13$.

But this is impossible, as the product of four distinct integers always has at least two (possibly non-distinct) prime factors.