(i) State and prove Rouché's theorem.

[You may assume the principle of the argument.]

(ii) Let $0<c<1$. Prove that the polynomial $p(z)=z^{3}+i c z+8$ has three roots with modulus less than 3. Prove that one root $\alpha$ satisfies $\operatorname{Re} \alpha>0, \operatorname{Im} \alpha>0$; another, $\beta$, satisfies $\operatorname{Re} \beta>0$, Im $\beta<0$; and the third, $\gamma$, has $\operatorname{Re} \gamma<0$.

(iii) For sufficiently small $c$, prove that $\operatorname{Im} \gamma>0$.

[You may use results from the course if you state them precisely.]