  Copi, Huterer, Starkman and Schwarz introduced multipole vectors in a tensor
context and used them to demonstrate that the first-year WMAP quadrupole and
octopole planes align at roughly the 99.9% confidence level. In the present
article the language of polynomials provides a new and independent derivation
of the multipole vector concept. Bezout's Theorem supports an elementary proof
that the multipole vectors exist and are unique (up to rescaling). The
constructive nature of the proof leads to a fast, practical algorithm for
computing multipole vectors. We illustrate the algorithm by finding exact
solutions for some simple toy examples, and numerical solutions for the
first-year WMAP quadrupole and octopole. We then apply our algorithm to Monte
Carlo skies to independently re-confirm the estimate that the WMAP quadrupole
and octopole planes align at the 99.9% level.
