How do I know if a simple pole exists (and how do I find it) for $f(z)$ without expanding the Laurent series first? In general, how do I recognize that a simple pole exists and find it, given some $\Large f(z)$? I want to do this without finding the Laurent series first.  
And specifically, for the following function: 
$\Large f(z) = \frac{z^2}{z^4+16}$

Here is how you find the roots of $z^4+16=0$,
$$ z^4 = -16 \Rightarrow z^4 = 16\, e^{i\pi} \Rightarrow z^4 = 16\, e^{i\pi+i2k\pi} \Rightarrow z = 2\, e^{\frac{i\pi + i2k\pi}{4} }\, k =0\,,1\,,2\,,3\,.$$ 
