  Let $F(x)$ be an irreducible polynomial with integer coefficients and degree
at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of
integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le
z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for
$y+y/\log^C y < z\le y^2$ and $y\le x^{1-\delta}$, showing that the order is
the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$
with a divisor in $(y,z]$. Here $C$ is an arbitrarily large constant and
$\delta>0$ is arbitrarily small.
