91Ó°ÊÓ

Express in the form \(a+b i\) : a. \(\frac{1}{6+2 i}\) b. \(\frac{(2+i)(3+2 i)}{1-i}\) c. \(\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{4}\) d. \(i^{2}, i^{3}, i^{4}, i^{5}, \ldots\)

Suppose \(P\) is a polynomial with real coefficients. Show that \(P(z)=0\) if and only if \(P(\bar{z})=0\) [i.e., zeroes of "real" polynomials come in conjugate pairs].

Verify that \(\left|z^{2}\right|=|z|^{2}\) using rectangular coordinates and then using polar coordinates.

Let \(z=x+i y .\) Explain the connection between \(\operatorname{Arg} z\) and \(\tan ^{-1}(y / x) .\) (Warning: they are not identical.)

Solve the following equations in polar form and locate the roots in the complex plane: a. \(z^{6}=1\). b. \(z^{4}=-1\). c. \(z^{4}=-1+\sqrt{3} i\).

Let Arg \(w\) denote that value of the argument between \(-\pi\) and \(\pi\) (inclusive). Show that $$ \operatorname{Arg}\left(\frac{z-1}{z+1}\right)=\left\\{\begin{array}{ll} \pi / 2 & \text { if } \operatorname{Im} z>0 \\ -\pi / 2 & \text { if } \operatorname{Im} z<0 \end{array}\right. $$ where \(z\) is a point on the unit circle \(|z|=1\).

Prove that \(x^{3}+p x=q\) has three real roots if and only if \(4 p^{3}<-27 q^{2}\). (Hint: Find the local minimum and local maximum values of \(\left.x^{3}+p x-q .\right)\)