Asymptotic magnitude Bode plot for a network function

[Cocoleia]

Homework Statement

I am working on the following problem:

The Attempt at a Solution

I have the solutions, I just don't understand:

I know that the pole should be the root of the denominator? but in all the examples I was given they used s and not jw, so what is the variable in the network function ? I am capable of deriving the network function, just a bit unsure of what the best method for finding the poles is.
I also have an example in the book where they say:

I've been trying to follow this logic. If we look at where ω<p1, I can see that they approximated both terms of the denominator by 1 and somehow got rid of the negative sign. For p1<ω<p2, it looks like they approximated the whole denominator by just jωC1R1...
I don't understand how or why they did what they did in the solutions. Thanks.

 

[jbsteele]

The answer to this problem is that the pole of the network function is the root of the denominator, which can be simplified using the following steps. For ω<p1, both terms of the denominator can be approximated by 1, as jωC1R1<<1 and jωC2R2<<1. Therefore, the denominator can be simplified to 1-1 = 0, and the pole is at p1 = 0.For p1<ω<p2, the denominator can be simplified as jω(C1R1+C2R2). Thus, the pole is at p2 = (C1R1+C2R2)^-1.Finally, for ω>p2, the denominator can be simplified as 1 + jω(C1R1+C2R2). Therefore, the pole is at p3 = -(C1R1+C2R2)^-1.