Let A be the algebra of quaternions H or octonions O. In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f(t) ∈ A[t] has a root in A. As a consequence, the Jacobian determinant |J(f)| is always nonnegative in A. Moreover, using the idea of the topological degree we show that a regular polynomial g(t) over A has also a root in A. Finally, utilizing multiplication (∗) in A, we prove various results on the topological degree of products of maps. In particular, if S is the unit sphere in A and h1, h2 : S → S are smooth maps, it is shown that deg(h1 ∗ h2) = deg(h1) + deg(h2).