  Multiplication of polynomials is among key operations in computer algebra
which plays important roles in developing techniques for other commonly used
polynomial operations such as division, evaluation/interpolation, and
factorization. In this work, we present formulas and techniques for polynomial
multiplications expressed in a variety of well-known polynomial bases without
any change of basis. In particular, we take into consideration degree-graded
polynomial bases including, but not limited to orthogonal polynomial bases and
non-degree-graded polynomial bases including the Bernstein and Lagrange bases.
All of the described polynomial multiplication formulas and techniques in this
work, which are mostly presented in matrix-vector forms, preserve the basis in
which the polynomials are given. Furthermore, using the results of direct
multiplication of polynomials, we devise techniques for intra-basis polynomial
division in the polynomial bases. A generalization of the well-known "long
division" algorithm to any degree-graded polynomial basis is also given. The
proposed framework deals with matrix-vector computations which often leads to
well-structured matrices. Finally, an application of the presented techniques
in constructing the Galerkin representation of polynomial multiplication
operators is illustrated for discretization of a linear elliptic problem with
stochastic coefficients.
