I have the expression
$$\frac{P(s)}{Q(s)}=\left(a_0~\sum\limits_{i=1}^n\frac{c_i}{s+\alpha_i}+1-a_0\right)\frac{B_1(s)}{\prod\limits_{i=1}^m s-\beta_i}$$ where $\alpha_i\neq\beta_i$ are all distinct roots and $deg(B_1(s))<m$. Using partial fraction method, I write
$$\frac{P(s)}{Q(s)}=\sum\limits_{i=1}^n\frac{D_i}{s+\alpha_i}+\sum\limits_{i=1}^m\frac{E_i}{s-\beta_i}$$ where $D_i=\frac{P(s)}{Q(s)'}|_{s=-\alpha_i}$ and $E_i=\frac{P(s)}{Q(s)'}|_{s=\beta_i}$ .
Simplifying the expression, I got $$D_k=\frac{a_0~c_k~B_1(-\alpha_k)}{\prod\limits_{j=1}^m(-\alpha_k-\beta_j)}$$ and $$E_k=\frac{\left(a_0~\sum\limits_{i=1}^n c_i\prod\limits_{j=1,j\neq i}^n(\beta_k+\alpha_j)+(1-a_0)\prod\limits_{j=1}^n \beta_k+\alpha_j\right)B_1(\beta_k)}{\prod\limits_{j=1}^n(\alpha_j+\beta_k)\prod\limits_{j=1,j\neq k}^m(\beta_k-\beta_j)}$$ Can any one please check the correctness of the coefficients and suggest some alternatives?