  In this work, multiplicities of zeros of general skew polynomials are
studied. Two distinct definitions are considered: First, $ a $ is said to be a
zero of $ F $ of multiplicity $ r $ if $ (x-a)^r $ divides $ F $ on the right;
second, $ a $ is said to be a zero of $ F $ of multiplicity $ r $ if some skew
polynomial $ P = (x-a_r) \cdots (x-a_2) (x-a_1) $, having $ a_1 = a $ as its
only right zero, divides $ F $ on the right. The first notion was considered
earlier, while the second one was recently introduced by Bolotnikov over the
quaternions. Neither of these two notions implies the other for general skew
polynomials. We show that, in the first case, Lam and Leroy's concept of
P-independence does not behave naturally, whereas a union theorem, as stated by
Lam, still holds. In contrast, we show that P-independence for the second
notion of multiplicities behaves naturally. As a consequence, we provide
extensions of classical commutative results to general skew polynomials. These
include the upper bound on the number of (P-independent) zeros (counting
multiplicities) of a skew polynomial by its degree, and the equivalence of
P-independence, the solvability of Hermite interpolation and the invertibility
of confluent Vandermonde matrices (for which we introduce skew polynomial Hasse
derivatives). We provide characterizations of skew polynomials of the form $ P
= (x-a_r) \cdots (x-a_2) (x-a_1) $ having a single right zero, assuming
conjugacy classes are algebraic. Based on these, we explicitly describe such
skew polynomials in particular cases of interest, recovering Bolotnikov's
results over the quaternions as a particular case.
