\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\i{\mathrm{i}}
\def\C{\mathbb{C}}
\def\N{\mathbb{N}}

\begin{document}

\title{Calculus and Elements of Linear Algebra I}
\author{Homework 1}
\date{Due on Moodle, September 14, 2020}
\maketitle

\begin{enumerate}

\item
\begin{enumerate}
\item
Find the (complex) roots of the polynomial
\[
  p(x) = x^2 + 4 \, x + 13 \,.
\]

\item Find the values of the parameter $\lambda$ for which the
equation
\[
  2 \, x^2 - \lambda \, x + \lambda = 0
\]
has no real solutions.
\end{enumerate}

\item Find all the roots (real or complex) of the polynomial
\[
  p(x) = 24 - 8 \, x - 18 \, x^2 + 18 \, x^3 - x^4 - 4 \, x^5 + x^6 \,.
\]
\emph{Hint:} $x=3$ is a root.  Divide out the associated linear factor
and continue with more roots that are easy to guess.

\item Assuming that $z = a + b \i$ is a complex number, compute real
and imaginary parts of
\begin{enumerate}
\item $\displaystyle \frac1{z^2}$,
\item $\displaystyle \frac{z+1}{2z-5}$,
\item $\displaystyle z^3$.
\end{enumerate}

\item \begin{enumerate}
\item Compute $\displaystyle \biggl\vert \frac{1+\i}{2-\i}
\biggr\vert$.
\item Characterize the set of real numbers $x$ that satisfy
\[
  \lvert 6 - 4 \, x \rvert \geq \lvert x-2 \rvert \,.
\]
\end{enumerate}

\item \begin{enumerate}
\item For $v, w \in \C$, show that $v^* \, w^* = (vw)^*$.
\item Conclude, inductively, that $(z^n)^* = (z^*)^n$ for $z\in \C$
and $n \in \N$.
\end{enumerate}

\end{enumerate}

\end{document}
