  Let $K$ be a field. Let $f\in K[[x_{1},...,x_{r}]]$ and $g\in
K[[y_{1},...,y_{s}]]$ be nonzero elements. If $X$ (resp. $Y$) is a matrix
factorization of $f$ (resp. $g$), Yoshino had constructed a tensor product (of
matrix factorizations) $\hat{\otimes}$ such that $X\hat{\otimes}Y$ is a matrix
factorization of $f+g\in K[[x_{1},...,x_{r},y_{1},...,y_{s}]]$. In this paper,
we propose a bifunctorial operation $\widetilde{\otimes}$ and its variant
$\widetilde{\otimes}'$ such that $X\widetilde{\otimes}Y$ and
$X\widetilde{\otimes}' Y$ are two different matrix factorizations of $fg\in
K[[x_{1},...,x_{r},y_{1},...,y_{s}]]$. We call $\widetilde{\otimes}$ the
multiplicative tensor product of $X$ and $Y$. Several properties of
$\widetilde{\otimes}$ are proved. Moreover, we find three functorial variants
of Yoshino's tensor product $\hat{\otimes}$. Then, $\widetilde{\otimes}$ (or
its variant) is used in conjunction with $\hat{\otimes}$ (or any of its
variants) to give an improved version of the standard algorithm for factoring
polynomials using matrices on the class of summand-reducible polynomials
defined in this paper. Our algorithm produces matrix factors whose size is at
most one half the size one obtains using the standard method.
