Why do we care about the multiplicity of poles and zeros in rational Z-transforms?

Question

In the DSP class I'm taking, a lot of the questions ask me to list the multiplicity of a pole or zero in a rational Z-transform. For example, the multiplicity of the zero at $z=1$ in :$\frac{(z-1)^2}{(z-5)}$ is $2$.

What knowledge do we gain by knowing the multiplicity of some pole or zero?

Of course, if we know the multiplicity of all the poles and zeros, we can describe the z-transform of a rational function, but this is not a very satisfying answer.

Answer 1

Multiplicity has important relevance when it comes to filter design. For example, Multiplicity will decide how sharp your attenuation is near zeros. Take two simple high pass filters as example $$ H_1(z) = 1-z^{-1}\\ H_2(z)=(1-z^{-1})^2=1-2z^{-1}+z^{-2} $$ The transfer function $H_1(z)$ has only one zero at $z=1$ but $H_2(z)$ has two zeros at $z=1$. Which means, for the fourier transform $H_2(e^{j\omega})$ will provide better attenuation at $\omega=0$

This is shown by a simple MATLAB example as below

h1=[1 -1];
h2 = [1 -2 1];
[H1,w]=freqz(h1);
[H2,w]=freqz(h2);
figure(1)
plot(w,20*log10(H1))
hold on
plot(w,20*log10(H2))
legend('H1','H2')

Answer 2

Multiple poles have amplitude and phase profile different to that of single poles.