Poles and zeros of a complex function

Good evening, I have to determine the poles and zeros of the following function with their respective multiplicity

$$f(z) = \frac{(\cos(z))^4\sin(z^3)}{z^6(z^3+2)}$$

I know that zeros are found by setting the numerator equal to zero while the poles are found by setting the denominator equal to zero.

For $(\cos(z))^4\sin(z^3)$ I have $z=π/2 +kπ $ for $k=0,1,2,\dots$ and $z=hπ$ for $h=0,1,2,3,\dots$

For $z^6(z^3+2)$ I have $z=0$ and $z=(-2)^\frac 13$

How do i find multiplicities?

Solution 1:

1. Zeros :
   1. $\dfrac{\pi}{2} + k \pi$ ($k \in \mathbb{N}$) with multiplicity $4$.
   2. $k \pi$ ($k \in \mathbb{N}^*$) with multiplicity $3$. $k$ must be $\neq 0$ coz $0$ is a pole ($6 > 3$).
2. Poles :
   1. $0$ with multiplicity $6 - 3 = 3$.
   2. $\sqrt[3]{2} \exp\left(\dfrac{i \pi}{3} + \dfrac{2 i k \pi}{3}\right)$ ($k \in \{0, 1, 2\}$ cubic roots of $-2$) with multiplicity $1$. They are simple.