\documentclass[a4paper]{article}
-\usepackage{standalone}
-\usepackage{newclude}
-\usepackage[a4paper,margin=2cm]{geometry}
-\usepackage{multicol}
-\usepackage{multirow}
+\usepackage[dvipsnames, table]{xcolor}
+\usepackage{adjustbox}
\usepackage{amsmath}
\usepackage{amssymb}
+\usepackage{blindtext}
+\usepackage{dblfloatfix}
+\usepackage{enumitem}
+\usepackage{fancyhdr}
+\usepackage[a4paper,margin=1.8cm]{geometry}
+\usepackage{graphicx}
\usepackage{harpoon}
+\usepackage{keystroke}
+\usepackage{listings}
+\usepackage{makecell}
+\usepackage{mathtools}
+\usepackage{mathtools}
+\usepackage{multicol}
+\usepackage{multirow}
+\usepackage{newclude}
+\usepackage{pgfplots}
+\usepackage{polynom}
+\usepackage{pst-plot}
+\usepackage{standalone}
+\usepackage{subfiles}
\usepackage{tabularx}
\usepackage{tabu}
-\usepackage{makecell}
-\usepackage[dvipsnames, table]{xcolor}
-\usepackage{blindtext}
-\usepackage{graphicx}
-\usepackage{wrapfig}
-\usepackage{tikz}
+\usepackage{tcolorbox}
\usepackage{tikz-3dplot}
-\usepackage{pgfplots}
-\pgfplotsset{compat=1.8}
-\usepackage{mathtools}
-\usetikzlibrary{calc}
-\usetikzlibrary{angles}
-\usetikzlibrary{datavisualization.formats.functions}
-\usetikzlibrary{decorations.markings}
+\usepackage{tikz}
+\usepackage{tkz-fct}
+\usepackage[obeyspaces]{url}
+\usepackage{wrapfig}
+
+
+\usetikzlibrary{%
+ angles,
+ arrows,
+ arrows.meta,
+ calc,
+ datavisualization.formats.functions,
+ decorations,
+ decorations.markings,
+ decorations.text,
+ decorations.pathreplacing,
+ decorations.text,
+ patterns,
+ scopes
+}
+
+\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
+
\usepgflibrary{arrows.meta}
-\usepackage{longtable}
-\usepackage{fancyhdr}
+\pgfplotsset{compat=1.16}
+\pgfplotsset{every axis/.append style={
+ axis x line=middle, % centre axes
+ axis y line=middle,
+ axis line style={->}, % arrows on axes
+ xlabel={$x$}, % axes labels
+ ylabel={$y$}
+}}
+
+\psset{dimen=monkey,fillstyle=solid,opacity=.5}
+\def\object{%
+ \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
+ \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90]
+ \rput{*0}{%
+ \psline{->}(0,-2)%
+ \uput[-90]{*0}(0,-2){$\vec{w}$}}
+}
+
\pagestyle{fancy}
\fancyhead[LO,LE]{Year 12 Methods}
\fancyhead[CO,CE]{Andrew Lorimer}
\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
+
+\newcommand{\tg}{\mathop{\mathrm{tg}}}
+\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
+\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
+\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
+
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
-\setlength{\parindent}{0cm}
-\usepackage{mathtools}
-\usepackage{xcolor} % used only to show the phantomed stuff
-\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
+\linespread{1.5}
+\setlength{\parindent}{0cm}
+\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
+
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\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}
+\newcolumntype{Y}{>{\centering\arraybackslash}X}
+
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-\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
-\pgfplotsset{every axis/.append style={
- axis x line=middle, % centre axes
- axis y line=middle,
- axis line style={->}, % arrows on axes
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-}}
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+\definecolor{lblue}{HTML}{e5e9f0}
+
+\newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm}
+\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=darkgray, fontupper=\sffamily\bfseries}
+\newtcolorbox{theorembox}[1]{colback=green!10!white, colframe=blue!20!white, coltitle=black, fontupper=\sffamily, fonttitle=\sffamily, #1}
+
\begin{document}
\begin{multicols}{2}
- \section{Functions}
- \begin{itemize}
- \tightlist
- \item vertical line test
- \item each \(x\) value produces only one \(y\) value
- \end{itemize}
+\section{Functions}
- \subsection*{One to one functions}
+\begin{itemize} \tightlist
+ \item vertical line test
+ \item each \(x\) value produces only one \(y\) value
+\end{itemize}
- \begin{itemize} \tightlist
- \item
- \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
- \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
- \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
- \(x^3\) is)
- \item
- horizontal line test
- \item
- if not one to one, it is many to one
- \end{itemize}
+\subsection*{One to one functions}
- \subsection*{Odd and even functions}
+\begin{itemize} \tightlist
+ \item \(f(x)\) is 1:1 if \(f(a) \ne f(b) \> \forall \>\{a,b\} \in \operatorname{dom}(f)\) \\
+ \(\implies\) unique \(y\) for each \(x\)
+ \item e.g. \(\sin x\) is not 1:1, \(x^3\) is
+ \item horizontal line test
+ \item if not one to one, it is many to one
+\end{itemize}
- \begin{align*}
- \text{Even:}&& f(x) &= f(-x) \\
- \text{Odd:} && -f(x) &= f(-x)
- \end{align*}
+\subsection*{Odd and even functions}
+
+\begin{align*}
+ \text{Even:}&& f(x) &= f(-x) \\
+ \text{Odd:} && -f(x) &= f(-x)
+\end{align*}
+
+Even \(\implies\) symmetrical across \(y\)-axis \\
+\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
+For \(x^n\), parity of \(n \equiv\) parity of function
+
+\begin{tabularx}{\columnwidth}{XX}
+ \textbf{Even:} & \textbf{Odd:} \\
+ \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} &
+ \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)}; \end{axis}\end{tikzpicture}
+\end{tabularx}
+
+\subsection*{Inverse functions}
+
+\begin{itemize} \tightlist
+ \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
+ \item \(f\) must be one to one
+ \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
+ \item Represents reflection across \(y=x\)
+ \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
+ \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
+ \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
+ \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
+\end{itemize}
- Even \(\implies\) symmetrical across \(y\)-axis \\
- \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
- For \(x^n\), parity of \(n \equiv\) parity of function
+\subsubsection*{Finding \(f^{-1}\)}
- \begin{tabularx}{\columnwidth}{XX}
- \textbf{Even:} & \textbf{Odd:} \\
- \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)}; \end{axis}\end{tikzpicture}
- \end{tabularx}
+\begin{enumerate} \tightlist
+ \item Let \(y=f(x)\)
+ \item Swap \(x\) and \(y\) (``take inverse''
+ \item Solve for \(y\) \\
+ Sqrt: state \(\pm\) solutions then restrict
+ \item State rule as \(f^{-1}(x)=\dots\)
+ \item For inverse \emph{function}, state in function notation
+\end{enumerate}
- \subsection*{Inverse functions}
+\subsection*{Simultaneous equations (linear)}
- \begin{itemize} \tightlist
- \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
- \item \(f\) must be one to one
- \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
- \item Represents reflection across \(y=x\)
- \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
- \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
- \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
- \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
- \end{itemize}
+\begin{itemize} \tightlist
+ \item \textbf{Unique solution} - lines intersect at point
+ \item \textbf{Infinitely many solutions} - lines are equal
+ \item \textbf{No solution} - lines are parallel
+\end{itemize}
- \subsubsection*{Finding \(f^{-1}\)}
+\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
+ where all coefficients are known except for one, and \(a, b\) are known
\begin{enumerate} \tightlist
- \item Let \(y=f(x)\)
- \item Swap \(x\) and \(y\) (``take inverse''
- \item Solve for \(y\) \\
- Sqrt: state \(\pm\) solutions then restrict
- \item State rule as \(f^{-1}(x)=\dots\)
- \item For inverse \emph{function}, state in function notation
+ \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
+ \item Find \(\det(\text{first matrix}) = ps-qr\)
+ \item Let \(\det = 0\) for \(\{0,\infty\}\) solutions
+ or \(\det \ne 0\) for 1 solution
+ \item Solve to find variable \\ \\
+ \textbf{For infinite/no solutions:}
+ \item Substitute variable into both original equations
+ \item Rearrange so that LHS of each is the same
+ \item \(\begin{aligned}[t]
+ \infty \text{ solns: } & \text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x \\
+ 0 \text{ solns: } & \text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x
+ \end{aligned}\)
\end{enumerate}
-
- \subsection*{Simultaneous equations (linear)}
-
- \begin{itemize} \tightlist
- \item \textbf{Unique solution} - lines intersect at point
- \item \textbf{Infinitely many solutions} - lines are equal
- \item \textbf{No solution} - lines are parallel
- \end{itemize}
-
- \subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
- where all coefficients are known except for one, and \(a, b\) are known
-
- \begin{enumerate} \tightlist
- \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
- \item Find determinant of first matrix: \(\Delta = ps-qr\)
- \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
- or let \(\Delta \ne 0\) for one unique solution.
- \item Solve determinant equation to find variable \\
- \textbf{For infinite/no solutions:}
- \item Substitute variable into both original equations
- \item Rearrange equations so that LHS of each is the same
- \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
- \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
- \end{enumerate}
-
- \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
-
- \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
-
- \begin{itemize} \tightlist
- \item Use elimination
- \item Generate two new equations with only two variables
- \item Rearrange \& solve
- \item Substitute one variable into another equation to find another variable
- \end{itemize}
-\subsection*{Piecewise functions}
+ \begin{cas}
+ Action \(\rightarrow\) Matrix \(\rightarrow\) Calculation \(\rightarrow\) \texttt{det}
+ \end{cas}
-\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
+ \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
-\textbf{Open circle:} point included\\
-\textbf{Closed circle:} point not included
+ \begin{itemize} \tightlist
+ \item Use elimination
+ \item Generate two new equations with only two variables
+ \item Rearrange \& solve
+ \item Substitute one variable into another equation to find another variable
+ \end{itemize}
-\subsection*{Operations on functions}
+ \subsection*{Piecewise functions}
-For \(f \pm g\) and \(f \times g\):
-\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
+ \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
-Addition of linear piecewise graphs: add \(y\)-values at key points
+ \textbf{Open circle:} point included\\
+ \textbf{Closed circle:} point not included
-Product functions:
+\begin{cas}
+ Define piecewise functions: \\
+ \-\hspace{1em}Math3 \(\rightarrow\)
+ \begin{tikzpicture}%
+ \draw rectangle (0.5,0.5);
+ \node at (0.08,0.25) {\(\{\)};
+ \filldraw [black] (0.15, 0.4) rectangle(0.25, 0.3);
+ \draw (0.35, 0.4) rectangle(0.45, 0.3);
+ \node [font=\footnotesize] at (0.3,0.3) {\verb;,;};
+ \draw (0.15, 0.2) rectangle(0.25, 0.1);
+ \node [font=\footnotesize] at (0.3,0.1) {\verb;,;};
+ \draw (0.35, 0.2) rectangle(0.45, 0.1);
+ \end{tikzpicture}
+ % TODO: finish this section
+\end{cas}
-\begin{itemize}
-\tightlist
-\item
- product will equal 0 if \(f=0\) or \(g=0\)
-\item
- \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
-\end{itemize}
+ \subsection*{Operations on functions}
-\subsection*{Composite functions}
+ For \(f \pm g\) and \(f \times g\):
+ \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
-\((f \circ g)(x)\) is defined iff
-\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
+ Addition of linear piecewise graphs: add \(y\)-values at key points
+ Product functions:
+
+ \begin{itemize}
+ \tightlist
+ \item
+ product will equal 0 if \(f=0\) or \(g=0\)
+ \item
+ \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
+ \end{itemize}
- \pgfplotsset{every axis/.append style={ ticks=none, xlabel=, ylabel=, }} % remove axis labels & ticks
- \begin{table*}[ht]
+ \subsection*{Composite functions}
+
+ \((f \circ g)(x)\) is defined iff
+ \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
+
+ \pgfplotsset{
+ blank/.append style={%
+ enlargelimits=true,
+ ticks=none,
+ yticklabels={,,}, xticklabels={,,},
+ xlabel=, ylabel=,
+ scale=0.4,
+ samples=100, smooth, unbounded coords=jump
+ }
+ }
+ \tikzset{
+ blankplot/.append style={orange, mark=none}
+ }
+
+ \begin{figure*}[ht]
\centering
- \begin{tabu} to \textwidth {@{} X[0.3,r] *2{|X[c,m]}@{}}
- & \(n\) is even & \(n\) is odd \\ \tabucline{1pt}
- \(x^n, n \in \mathbb{Z}^+\) &
- \vspace{1em}\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^3)}; \end{axis}\end{tikzpicture} \\
- \(x^n, n \in \mathbb{Z}^-\) &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-4, xmax=4, ymax=8, ymin=-0, scale=0.4, smooth] \addplot[orange, mark=none, samples=100] {(x^(-2))}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none, domain=-3:-0.1] {(x^(-1))}; \addplot[orange, mark=none, domain=0.1:3] {(x^(-1))}; \end{axis}\end{tikzpicture} \\
- \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-1, xmax=5, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^(1/2))}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}
- \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4]
- \addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
- \end{axis}
- \end{tikzpicture}
- \end{tabu}
- \hrule
- \end{table*}
- \pgfplotsset{every axis/.append style={ xlabel=\(x\), ylabel=\(y\) }} % put axis labels back
+
+ \begin{tabularx}{\textwidth}{|r|Y|Y|}
+
+ \hline
+ \rowcolor{lblue}
+ & \(n\) is even & \(n\) is odd \\ \hline
+
+ \centering \(x^n, n \in \mathbb{Z}^+\) &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3]
+ \addplot[blankplot] {(x^2)};
+ \end{axis}
+ \end{tikzpicture}} &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3]
+ \addplot[blankplot, domain=-3:3] {(x^3)};
+ \end{axis}
+ \end{tikzpicture}} \\ \hline
+
+ \centering \(x^n, n \in \mathbb{Z}^-\) &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
+ \addplot[blankplot, samples=100] {(x^(-2))};
+ \end{axis}
+ \end{tikzpicture}} &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3]
+ \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
+ \addplot[blankplot, domain=0.1:3] {(x^(-1))};
+ \end{axis}
+ \end{tikzpicture}} \\ \hline
+
+ \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-1, xmax=5]
+ \addplot[blankplot] {(x^(1/2))};
+ \end{axis}
+ \end{tikzpicture}} &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
+ \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
+ \end{axis}
+ \end{tikzpicture}} \\ \hline
+
+ \end{tabularx}
+ \end{figure*}
\section{Polynomials}
+ \subsection*{Factor theorem}
+
+ \begin{theorembox}{title=General form \(\beta x + \alpha\)}
+ If \(\beta x + \alpha\) is a factor of \(P(x)\), \\
+ \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
+ \end{theorembox}
+
+ \begin{theorembox}{title=Simple form \(x-a\)}
+ If \((x-a)\) is a factor of \(P(x)\), remainder \(R=0\). \\
+ \-\hspace{1em}\(\implies P(a)=0\)
+ \end{theorembox}
+
+ \subsection*{Remainder theorem}
+
+ \begin{theorembox}{}
+ When \(P(x)\) is divided by \(\beta x + \alpha\), the remainder is \(-\dfrac{\alpha}{\beta}\).
+ \end{theorembox}
+
+ \subsection*{Rational root theorem}
+ Let \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) be a polynomial of degree \(n\) with \(a_i \in \mathbb{Z} \forall a\). Let \(\alpha, \beta \in \mathbb{Z}\) such that their highest common factor is 1 (i.e. relatively prime).
+
+ If \(\beta x + \alpha\) is a factor of \(P(x)\), then \(\beta\) divides \(a_n\) and \(\alpha\) divides \(a_0\) .
+
+ \subsubsection*{Discriminant}
+ \[\begin{cases}
+ b^2-4ac > 0 & \text{two solutions} \\
+ b^2-4ac = 0 & \text{one solution} \\
+ b^2-4ac < 0 & \text{no solutions}
+ \end{cases}\]
+ \begin{warning}
+ Flip inequality sign when multiplying by -1
+ \end{warning}
+
+ \subsection*{Long division}
+
+ \[ \polylongdiv{x^2+2x+4}{x-1} \]
+
+ \begin{cas}
+ Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{propFrac}
+ \end{cas}
+
\subsection*{Linear equations}
\subsubsection*{Forms}
Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
\subsection*{Quadratics}
+
\setlength{\abovedisplayskip}{1pt}
\setlength{\belowdisplayskip}{1pt}
+
+ \textbf{Linear factorisation}
\[ x^2 + bx + c = (x+m)(x+n) \]
\hfill where \(mn=c, \> m+n=b\)
\input{circ-functions}
\input{calculus}
+ \subfile{statistics-ref}
+
\end{multicols}
- \end{document}
+
+\end{document}