write all functions as rational functions

Question

my question is short and simple : could we write all functions that have the formula : $1 + G(s)H(s)$ as

$$\frac{(s+b_0)(s+b_1)\dots(s+b_n)}{(s+p_0)\dots(s+p_m)}$$

if the answer is yes , could you proof it ? B.S : this formula is famouse in control theory or control systems .

Answer 1

Of course, because $G(s)$ and $H(s)$ are both rational polynomials. Let $G(s)=N_G(s) / D_G(s)$ and $H(s)=N_H(s) / D_H(s)$ where $N_G(s)$, $D_G(s)$, $N_H(s)$ and $D_H(s)$ are polynomials. Then,

$$1 + G(s) H(s) = \frac{D_G(s) D_H(s) + N_G(s) N_H(s)}{D_G(s) D_H(s)} $$