Let $\mathbb{T}=\{z:|z|=1\}$ be the unit circle in $\mathbb{C}$, and let $\phi: \mathbb{T} \rightarrow \mathbb{C}$ be a continuous function that never takes the value 0 . Define the degree (or winding number) of $\phi$ about 0 . [You need not prove that the degree is well-defined.]

Denote the degree of $\phi$ about 0 by $w(\phi)$. Prove the following facts.

(i) If $\phi_{1}$ and $\phi_{2}$ are two functions with the properties of $\phi$ above, then $w\left(\phi_{1} \cdot \phi_{2}\right)=$ $w\left(\phi_{1}\right)+w\left(\phi_{2}\right)$

(ii) If $\psi$ is any continuous function such that $|\psi(z)|<|\phi(z)|$ for every $z \in \mathbb{T}$, then $w(\phi+\psi)=w(\phi)$.

Using these facts, calculate the degree $w(\phi)$ when $\phi$ is given by the formula $\phi(z)=$ $(3 z-2)(z-3)(2 z+1)+1 .$