Let $\mathbb P$ be the set of prime numbers.

Is there a non constant polynomial $f \in \mathbb Z[X]$ such that the set $$I_f := \{ \textstyle\frac{z}{p} : z \in \mathbb Z, p \in \mathbb P, p \mid f(z) \}$$ is dense in $\mathbb R$?

(I suppose that the following much stronger statement holds: Every polynomial $f \in \mathbb Z[X]$ which has a irreducible factor of degree at least $2$ satisfies the above condition.)