  We provide upper bounds for the sum of the multiplicities of the non-constant
irreducible factors that appear in the canonical decomposition of a polynomial
$f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius
circle associated to two points on the real axis with integer abscissae $a$ and
$b$, with ratio of the distances to these points depending on the admissible
divisors of $f(a)$ and $f(b)$. In particular, we obtain such upper bounds for
the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is an
Enestr\"om-Kakeya polynomial, or a Littlewood polynomial, or has a large
leading coefficient. Similar results are also obtained for multivariate
polynomials over arbitrary fields, in a non-Archimedean setting.
