Given a field $K$, $r$ matrices $D_i \in K^{n \times n}$, a matrix $M \in K^{n \times m}$ of rank at most $r$, in this paper, we study the problem of factoring $M$ as follows $M=\sum_{i=1}^r D_i \, u \, v_i$, where $u \in K^{n \times 1}$ and $v_i \in K^{1 \times m}$ for $i=1, \ldots, r$. This problem arises in modulation-based mechanical models studied in gearbox vibration analysis (e.g., amplitude and phase modulation). We show how linear algebra methods combined with linear system theory ideas can be used to characterize when this polynomial problem is solvable and if so, how to explicitly compute the solutions.