In general, if we are faced with a rational function, is the order of the pole always determined by the power to which the pole is raised?
FOr example,
$$ f(z) = \frac{z^2+1}{(z-2)(z-5)z^2}$$
Do we know that $z=0$ is a pole of order $0$? Or must we check that it is not an essential singularity or a removatlbe singularity by expanding into a Laurent Series?
I wonder if this is true for for rational functions. I can find counter examples for other functions, but not for rational functions.