  The purpose of this paper is to show how the problem of finding the zeros of
unilateral n-order quaternionic polynomials can be solved by determining the
eigen-vectors of the corresponding companion matrix. This approach, probably
superfluous in the case of quadratic equations for which a closed formula can
be given, becomes truly useful for (unilateral) n-order polynomials. To
understand the strehgth of this method, we compare it with the Niven algorithm
and show where this (full) matrix approach improves previous methods based on
the use of the Niven algorithm. For the convenience of the readers, we
explicitly solve some examples of second and third order unilateral
quaternionic polynomials. The leading idea of the practical solution method
proposed in this work can be summarized in following three steps: translating
the quaternionic polynomial in the eigenvalue problem for its companion matrix,
finding its eigenvectors, and, finally, giving the quaternionic solution of the
unilateral polynomial in terms of the components of such eigenvectors. A brief
discussion on bilateral quaternionic quadratic equations is also presented.
