  Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root
$\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ for some $g(x) \in A[x]$. 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 $C_f(x)=f(x)\cdot \overline{f(x)}$ are studied. Finally, we compute
the left eigenvalues of $2\times 2$ octonion matrices.
