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Just wanted to make sure:

This cases would have three poles at the exact same location of (z=1) on the complex plane?

$H(z)=\frac{1}{(z-1)^3}$

But this case, would have three poles spread out on a circle of complex plane:

$H(z)=\frac{1}{z^3-1}$

$H(z)=\frac{1}{\left(z-1\right)\left(z-0.5-i\sqrt3/2\right)\left(z-0.5+i\sqrt3/2\right)}$

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Exactly. In the first case you have all 3 poles $z_{1,2,3}=1$ in a real part of the complex plane. enter image description here

In the second one you have the poles spread on the circle as you suggested

$z_1 = 1$ , $z_2 = - \frac{1}{2} + \frac{1}{2}\cdot\sqrt{3}i$, $z_3 = - \frac{1}{2} - \frac{1}{2}\cdot\sqrt{3}i$

enter image description here

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