Some equations over generalized quaternion and octonion division algebras

It is known that any polynomial of degree n with coefficients in a field K has at most n roots in K. If the coefficients are in H (the quaternion algebra), the situation is different. For H over the real field, there is a kind of a fundamental theorem of algebra If it polynomial has only one term of the greatest degree then it hits at least one root in H. A similar theorem is also true for the octonions In this paper we try to solve, in general or in particular cases, some quadratic and linear equations with two different terms of greatest degree and the coefficients in the generalized division quaternion and octonion algebras H(alpha, beta) and O(alpha beta)over an arbitrary field K. char K not equal 2