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.

• 17A35 - Nonassociative division algebras
• 17A45 - Quadratic algebras (but not quadratic Jordan algebras)
• 17A75 - Composition algebras
• 17D05 - Alternative rings
• 37C25 - Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
• 37P35 - Arithmetic properties of periodic points