For any given matrix $F(z)\in M_{l\times m}(R)$ where $M_{l\times m}(R)$ is the set of $l\times m$ matrices with entries in $R= K[z_1,\ldots,z_n]$ and $l\leq m$, let $D_l(F)$ to be the greatest common divisor of all $l \times l$ minors, and assume that $d_0(z)$ has the form of $z_1 - f(z_2,\ldots,z_n)$. Suppose $(d_0(z))^r\mid D_l(F)$, this work is to research how to factorize $F(z)$ into the product of $G_0(z)\in M_{l\times l}(R)$ and $F_0(z)\in M_{l\times m}(R)$ such that $det(G_0) = (d_0(z))^{r_0}$, where $1\leq r_0\leq r$. The main ideal in our paper is to extract more than one common divisor $d_0(z)$ from $D_l(F)$ each time when $F(z)$ is factorized, this conclusion is a generalization of theorem in Lin et al.(2001). Moreover, we do an analysis of the condition that $F(z)$ must satisfy while using the above result. As a consequence, we obtain a generalized algorithm for our method. Compared to the previous results in Liu et al.(2011), the application range of our algorithm is more extensive. First we use Lin-Bose Theorem to construct a zero left prime matrix $H(z)\in M_{r_0\times l}(R)$ such that $H(z)F(z) = 0_{r_0\times m}$, where ${\rm Rank}(F(z))=l-r_0$ for each $z\in K^n$ and $1\leq r_0 \leq l$. Meanwhile, the famous Quillen-Suslin Theorem is used to construct a square n-D polynomial matrix $U(z)\in M_{l\times l}(R)$ such that $det(U) = c$ with $c$ is a nonzero constant and $H(z)$ is the first $r_0$ rows of $U(z)$. Then $F(z)$ admits a matrix factorization with respect to $(d_0(z))^{r_0}$: $F(z)=G_0(z)F_0(z)$, where $G_0(z) = U^{-1}(z)\cdot diag\{d_0(z),\ldots,d_0(z),1,\ldots,1\}$. We now turn to analyze the necessary and sufficient condition when ${\rm Rank}(F(z))=l-r_0$ for each $z\in K^n$. Inspired by the theorem in Liu et al.(2011), we get the following theorem: Let $F(z)\in M_{l\times m}(R)$ with a full row rank, assume that $(d_0(z))^r|D_l(F)$ and $1\leq r_0 \leq min\{l,r\}$, then $\langle d_0(z),c_1^{l-r_0}(z),\ldots,c_{\eta_{l-r_0}}^{l-r_0}$ $(z)\rangle=R$ and $d_0(z)\mid D_{l-r_0+1}(F)$ if and only if ${\rm Rank}(F(f,z_2,\ldots,z_n))=l-r_0$ for each $(z_2,\ldots,z_n)$ $\in K^{n-1}$, where $c_1^{l-r_0}(z),\ldots,c_{\eta_{l-r_0}}^{l-r_0}(z)$ denote the all $(l-r_0)\times (l-r_0)$ minors of matrix $F(z)$. According to the above two results, we get an algorithm and three nontrivial examples are given to show the effectiveness of this algorithm.