  We study three convolutions of polynomials in the context of free probability
theory. We prove that these convolutions can be written as the expected
characteristic polynomials of sums and products of unitarily invariant random
matrices. The symmetric additive and multiplicative convolutions were
introduced by Walsh and Szeg\"o in different contexts, and have been studied
for a century. The asymmetric additive convolution, and the connection of all
of them with random matrices, is new. By developing the analogy with free
probability, we prove that these convolutions produce real rooted polynomials
and provide strong bounds on the locations of the roots of these polynomials.
