I have designed an active inverting bandpass filter, and when comparing its pole zero plot to a passive bandpass fitler with the same cutoff frequencys, the pole zero plot is exactly the same, even though the bode plots for both are different. I will run through my calculations and results below. 
[![enter image description here](https://i.stack.imgur.com/cDZzB.jpg)](https://i.stack.imgur.com/cDZzB.jpg)

With a gain of 2.5, Lower cutoff of 75Hz, and uppper of 31kHz, I found component values to be: \$R\_1 = 2122\Omega\$, \$R\_2 = 5305\Omega\$, \$C\_1 = 1\mu\text{F}\$, \$C\_2 = 0.967\text{nF}\$. From which I derived the transfer function and came find the following:

$$H(s) = -\frac{R\_2C\_1s}{(1+C\_1R\_1s)(1+C\_2R\_2s)}$$

$$=> H(s) = -\frac{5.305\times 10^{-3}s}{1.088\times 10^{-8}s^2 + 2.127\times 10^{-3}s + 1}$$

Using Matlab, I got the following result (Bode and Pole Zero Map)
[![enter image description here](https://i.stack.imgur.com/PE9TT.jpg)](https://i.stack.imgur.com/PE9TT.jpg)

[![enter image description here](https://i.stack.imgur.com/Z3N1S.jpg)](https://i.stack.imgur.com/Z3N1S.jpg)

To compare how the system responded compared to a passive bandpass filter seen below, I used derived the transfer function and placed the same values used for the other filter, thus giving the same cut-off frequencies.

[![enter image description here](https://i.stack.imgur.com/n2fEL.jpg)](https://i.stack.imgur.com/n2fEL.jpg)

$$H(s) = \frac{R\_2C\_2s}{(1+R\_2C\_2s)(1+R\_1C\_1s+ \frac{R\_2}{R\_1}R\_1C\_1s)}$$

$$=> H(s) = \frac{5.13\times 10^{-6}s}{3.81\times 10^{-8}s^2 + 7.432\times 10^{-3}s + 1}$$

When processing this in MATLAB, I get the following plots:

[![enter image description here](https://i.stack.imgur.com/wnQp2.jpg)](https://i.stack.imgur.com/wnQp2.jpg)
[![enter image description here](https://i.stack.imgur.com/Nqa60.jpg)](https://i.stack.imgur.com/Nqa60.jpg)

What is didn't expect or understand is the pole zero response. What do the poles and zeros represent in this system, and also shouldn't the opamp introduce its own poles and zeros and its is adding gain to the system. I tried reading on the pole zeros responses of low pass and high pass filters on google, and they look nothing like my results! Added to that, I was originally trying to understand at what point of the system the gain is applied too, from looking at this response, is it correct to assume the system is LowPass>Gain>HighPass?