Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras

Over a composition algebra A, a polynomial f(x) ∈ A[x] has a root α if and only f(x)=g(x)· (x-α) for some g(x) ∈ A[x]. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that it is when $f(x)$ is linear or monic quadratic, but it is false in general. Similar questions about the connections between f and its companion Cf(x)=f(x)· f(x)̄ are studied. Finally, we compute the left eigenvalues of 2× 2 octonion matrices.