  A sum of a large-dimensional random matrix polynomial and a fixed low-rank
matrix polynomial is considered. The main assumption is that the resolvent of
the random polynomial converges to some deterministic limit. A formula for the
limit of the resolvent of the sum is derived and the eigenvalues are localised.
Three instances are considered: a low-rank matrix perturbed by the Wigner
matrix, a product $HX$ of a fixed diagonal matrix $H$ and the Wigner matrix $X$
and a special matrix polynomial. The results are illustrated with various
examples and numerical simulations.
