  We define \emph{generalized standard triples} $\mathbf{X}$, $\mathbf{Y}$, and
$L(z) = z\mathbf{C}_{1} - \mathbf{C}_{0}$, where $L(z)$ is a linearization of a
regular matrix polynomial $\mathbf{P}(z) \in \mathbb{C}^{n \times n}[z]$, in
order to use the representation $\mathbf{X}(z
\mathbf{C}_{1}~-~\mathbf{C}_{0})^{-1}\mathbf{Y}~=~\mathbf{P}^{-1}(z)$ which
holds except when $z$ is an eigenvalue of $\mathbf{P}$. This representation can
be used in constructing so-called \emph{algebraic linearizations} for matrix
polynomials of the form $\mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) +
\mathbf{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of
$\mathbf{A}(z)$ and $\mathbf{B}(z)$. This can be done even if $\mathbf{A}(z)$
and $\mathbf{B}(z)$ are expressed in differing polynomial bases. Our main
theorem is that $\mathbf{X}$ can be expressed using the coefficients of the
expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant
polynomial basis. For convenience, we tabulate generalized standard triples for
orthogonal polynomial bases, the monomial basis, and Newton interpolational
bases; for the Bernstein basis; for Lagrange interpolational bases; and for
Hermite interpolational bases. We account for the possibility of common
similarity transformations.
