Zeros of quaternion polynomials

It is well known that, over a division ring, every zero of a polynomial f(x ) = (x - x(1)) (...) (x - x(n)) is congruent to x(r) for some r. In this no te, we show further that, over the quaternion field, there exists at least one quaternion q(r) congruent to each x(r), and that, through this result, a constructive method for determining the zeros of quaternion polynomials c an be established. (C) 2000 Elsevier Science Ltd. All rights reserved.