Roots of polynomials over division rings

In this article, we study the properties of polynomials over division rings. Formulas for finding roots of polynomials which are the products of linear factors are obtained. These formulas generalize the known results for quaternion algebras. As known, if a minimal polynomial of a conjugacy class A in a noncommutative division ring is quadratic, then any polynomial having two roots in A vanishes identically on A. We show that in the case of a conjugacy class with minimal polynomial of larger degree, the situation is completely different. For any conjugacy class with minimal polynomial of degree >2, we construct a quadratic polynomial with infinitely many roots in this class, but there also are infinitely many elements in this class which are not the roots of this polynomial.