I'm working with theory of analog filters, typically the building blocks of these devices are the ratio of two polynomials (transfer function) $$ H(s)=\frac{G[s^{m}+a_{m-1}s^{m-1}+a_{m-2}s^{m-2}+...a_{1}s+a_{0}]}{[s^{n}+b_{n-1}\text{⋅}s^{n-1}+b_{n-2}\text{⋅}s^{n-2}+...+b_{1}\text{⋅}s+b_{0}]} $$ In this representation, the numerator and denominator polynomials usually are separated into first-order factors to ease the physical implementation.
$$ H(s)=\frac{G[(s+z_{'0})(s+z_{1})...(s+z_{m-2})(s+z_{m-1})]}{[(s+p_{'0})(s+p_{1})...(s+p_{m-2})(s+p_{m-1})]} $$
Particullary the interesting ones for me are ratios in the next form $$ H(s)=\frac{1}{s^{2}+\alpha s+1} $$
Now for a certain filter I have especified the next function $$ H(s)=\frac{1}{(\frac{s}{1013.603854}+1)((\frac{s}{1013.603854})^{2}+0.6180(\frac{s}{1013.603854})+1)((\frac{s}{1013.603854})^{2}+1.6180(\frac{s}{1013.603854})+1)} $$
But these needs to be normalizated, and order to do so I multiply the ratio in the next way $$ H(s)=\frac{1}{\frac{s}{1013.603854}+1}(\frac{1013.603854}{1013.603854})\frac{1}{(\frac{s}{1013.603854})^{2}+0.6180(\frac{s}{1013.603854})+1)}(\frac{1013.603854}{1013.603854})^{2}\frac{1}{(\frac{s}{1013.603854})^{2}+1.6180(\frac{s}{1013.603854})+1}(\frac{1013.603854}{1013.603854})^{2} $$
This yields the next result $$ H(s)=\frac{(1013.603854)^{5}}{(s+1013.603854)(s^{2}+626.407181772s+1013.603854^{2})(s^{2}+1640.011035772s+1013.603854{}^{2})} $$ It seems to be fine, but I'm not aware if these is the right way to simplify since it seems I'm not applying the distributive property in the denominator: $$ (P\land(Q\lor R))\Leftrightarrow((P\land Q)\lor(P\land R)) $$
Finally,
- Is this the right method to simplify?(a.k.a. is this operation distributive?)
Perhaps I'm overthinking, but it seems not to conform the definition of left and right distributive binary operations.