  We show how to construct linearizations of matrix polynomials
$z\mathbf{a}(z)\mathbf{d}_0 + \mathbf{c}_0$, $\mathbf{a}(z)\mathbf{b}(z)$,
$\mathbf{a}(z) + \mathbf{b}(z)$ (when $\mathrm{deg}\left(\mathbf{b}(z)\right) <
\mathrm{deg}\left(\mathbf{a}(z)\right)$), and
$z\mathbf{a}(z)\mathbf{d}_0\mathbf{b}(z) + \mathbf{c_0}$ from linearizations of
the component parts, $\mathbf{a}(z)$ and $\mathbf{b}(z)$. This allows the
extension to matrix polynomials of a new companion matrix construction.
