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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 $g(x) \in A[x]$. Weexamine whether this is true for general Cayley-Dickson algebras. Theconclusion is that it is when $f(x)$ is linear or monic quadratic, but falseotherwise. Similar questions about the connections between $f$ and itscompanion $C_f(x)=f(x)\cdot \overline{f(x)}$ are studied. Finally, we computethe left eigenvalues of $2\times 2$ octonion matrices.

Roots and Right Factors of Polynomials and Left Eigenvalues of Matrices over Cayley-Dickson Algebras