  It is known that polynomials over quaternions may have spherical zeros and
isolated left and right zeros. These zeros along with appropriately defined
multiplicities form the zero structure of a polynomial. In this paper, we
equivalently describe the zero structure of a polynomial in terms of its left
and right spherical divisors as well as in terms of left and right
indecomposable divisors. Several algorithms are proposed to find left/right
zeros and left/right spherical divisors of a quaternion polynomial, to
construct a polynomial with prescribed zero structure and more generally, to
construct the least left/right common multiple of given polynomials. Similar
questions are briefly discussed in the setting of quaternion formal power
series.
