Suppose $m$ is a positive integer. Define a polynomial $x^{2m}+1$ and denote $x_0,x_1,...,x_{2m}$ as zeros of $x^{2m}+1$, where $x_{j+m}=\bar{x_j}$, $1 \leq j \leq m$.
If matrices $A$ and $B$ commute, then
$$A^{2m+1} + B^{2m+1} = (A+B)\prod_{j=1}^{m}{(A-x_jB)}{(A-\bar{x_j}B)}$$
Question: Why the equality above holds? I try to see what's happening by letting $m=1$.