\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{harpoon}
+\usepackage{tabularx}
\usepackage{graphicx}
\usepackage{wrapfig}
+\usepackage{tikz}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead[LO,LE]{Year 12 Specialist}
\fancyhead[CO,CE]{Andrew Lorimer}
+
+\usepackage{mathtools}
+\usepackage{xcolor} % used only to show the phantomed stuff
+\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
+\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
+\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
+\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
+
\begin{document}
\begin{multicols}{2}
& \qquad \text{(rationalise denominator)}
\end{align*}
+ \subsection*{Argand planes}
+
+ \begin{tikzpicture}\begin{scope}[thick,font=\scriptsize]
+ \draw [->] (-1.5,0) -- (1.5,0) node [above left] {$\operatorname{Re}(z)$};
+ \draw [->] (0,-1.5) -- (0,1.5) node [below right] {$\operatorname{Im}(z)$};
+
+ % If you only want a single label per axis side:
+ \draw (1,-3pt) -- (1,0pt) node [below] {$1$};
+ \draw (-1,-3pt) -- (-1,0pt) node [below] {$-1$};
+ \draw (-3pt,1) -- (0pt,1) node [left] {$i$};
+ \draw (-3pt,-1) -- (0pt,-1) node [left] {$-i$};
+ \end{scope}\end{tikzpicture}
+
+ Multiplication by \(i \implies\) anticlockwise rotation of \(\frac{\pi}{2}\)
+
+ \subsection*{de Moivres' theorem}
+
+ \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
+
+ \subsection*{Complex polynomials}
+
+ Include \(\pm\) for all solutions, incl. imaginary
+
+\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
+\newcolumntype{L}{>{\raggedright\arraybackslash}X}
+ \begin{tabularx}{\columnwidth}{rX}
+ \hline
+ Sum of squares & \(\begin{aligned}
+ z^2 + a^2 &= z^2-(ai)^2\\
+ &= (z+ai)(z-ai) \end{aligned}\) \\
+ \hline
+ Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
+ \hline
+ Division & \(P(z)=D(z)Q(z)+R(z)\) \\
+ \hline
+ \parbox[t]{2cm}{Remainder} & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
+ \hline
+\end{tabularx}
+
+\subsection*{Roots}
+
+\(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
+
+\[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
+
+\begin{itemize}
+
+ \item{Same modulus for all solutions}
+ \item{Arguments are separated by \(\frac{2\pi}{n}\)}
+
+\item{Solutions of \(z^n=a\) where \(a \in \mathbb{C}\) lie on the circle \(x^2+y^2=\left(|a|^{\frac{1}{n}}\right)^2\)}
+\end{itemize}
+
+\subsubsection*{Conjugate root theorem}
+
+If \(a+bi\) is a solution to \(P(z)=0\), then the conjugate \(\overline{z}=a-bi\) is also a solution.
+
\end{multicols}
\end{document}