A factorisation worksheet tracing a straight line from simple common-factor stuff to quadratic trinomials.

factorisation.tex

\documentclass[10pt,a4paper]{article}

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% Copyright (c) 2019 Gavin Sinclair, released under the Do What You Want With It licence.
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\lfoot{\small\itshape Factorisation}
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%    Commands section: question, q, qw, qww, qwww, qhww
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% Counter for the 'part' of a question: (a) (b) (c) etc.
\newcounter{gspart}

% e.g. \question{3}{Simplify:}
%        \qw{2a+7-3a-9}
%        \qww{4(p-3}-(p-1)}
\newcommand{\question}[2]{
  \needspace{1.5\baselineskip}
  \setcounter{gspart}{0}
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    \textbf{Question #1}~~#2
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% \q Creates a one-line question (no lines for working or answer): \qw{3+5}
% Intended as a building block, but may be useful as is.
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\newcommand{\q}[1]{\
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   \tab\tab\tab $=$ \tab \underline{\hspace{2in}}\hspace{-2in}#1 }

% \qw Creates a two-line question: \qw{4x-5+3x+7}
\newcommand{\qw}[1]{
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}

% \qww Creates a three-line question: \qww{3x(x+7)+2(x-5)}
\newcommand{\qww}[1]{
  \begin{minipage}[t]{\columnwidth}
    \q{#1} \\
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}

% \qwww Creates a four-line question: \qwww{2ac+5bc-4ad-10bd}
\newcommand{\qwww}[1]{
  \begin{minipage}[t]{\columnwidth}
    \q{#1} \\
    \workingline \\
    \workingline \\
    \workingline \\
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}

% \qhww Creates a four-line question where the first working line is filled in:
%    e.g. \qhww{x^2+5x+6}{x^2+3x+2x+6}
\newcommand{\qhww}[2]{
  \begin{minipage}[t]{\columnwidth}
    \q{#1} \\
    \workinglineInsert{$#2$} \\
    \workingline \\
    \workingline \\
  \end{minipage}
}

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%    Beginning of document
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\begin{document}

\begin{multicols*}{2}
  [\begin{center}
    {\LARGE Factorisation worksheet}\\
    \vspace{1cm}
  \end{center}]

  \question{1}{Factorise:}
    \qw{4x+10}
    \qw{4x-10}
    \qw{-4x+10}
    \qw{-4x-10}
    \qw{x^3+4x^2+2x}
    \qw{-15a-3a^2}
    \qw{3mn^3+9mn}
    \qw{-4x^2y+2xy}
    \qw{3q^5-2q^2}
    \qw{7a^3b+5a^2b^2+4a^2b^3}

  \question{2}{Factorise:}
    \qw{3a(a-4)+2(a-4)}
    \qw{x(m+5)+7(m+5)}
    \qw{5a^2(b-c)-2y(b-c)}
    \qw{2(y-1)-y(y-1)}
    \qw{3x(x+2)+5(x+2)}
    \qw{3x(x+2)+(x+2)}
    \qw{3x(x+2)-(x+2)}
    \qw{a(3m-1)+b(3m-1)-c(3m-1)}
    \qww{4p(2a-1)+3(1-2a)}
    \qww{m(x-3)-8(3-x)}                
    \qwww{(x-3)^2+5(x-3)}
    \qwww{a(a+1)-(a+1)^2}
    \qwww{a(a+1)-4(a+1)^2}

  \question{3}{Factorise by grouping in pairs:}
    \qww{7x+14y+bx+2by}
    \qww{6x+9+2ax+3a}
    \qww{mp+mq+np+nq}
    \qww{ax-ay+bx-by}
    \qww{ax+3a+2x+6}
    \qww{a^2+ab+ac+bc}
    \qww{z^3-z^2+z-1}
    \qww{ac+bc-ad-bd}
    \qww{pu-qu-pv+qv}
    \qww{x^2-3x-xy+3y}
    %\qww{5p-5q-px+qx}
    %\qww{2ax-bx-2ay+by}
    \qww{ab+ac-b-c}
    \qww{x^3+4x^2-3x-12}
    \qww{a^3-3a^2-2a+6}
    \qww{2t^3+5t^2-10t-25}
    \qww{2x^3-6x^2-ax+3a}

  \question{4}{Factorise these quadratic trinomials, taking special note of the first step:}
    \qhww{3x^2+4x+1}{3x^2+3x+x+1}
    \qhww{2x^2+5x+2}{2x^2+4x+x+2}
    \qhww{3x^2+16x+5}{3x^2+15x+x+5}
    \qhww{3x^2+8x+4}{3x^2+6x+2x+4}
    \qhww{2x^2-3x+1}{2x^2-2x-x+1}
    \qhww{5x^2-13x+6}{5x^2-10x-3x+6}
    \qhww{5x^2-11x+6}{5x^2-5x-6x+6}
    \qhww{6x^2-11x+3}{6x^2-9x-2x+3}
    \qhww{2x^2-x-3}{2x^2-3x+2x-3}
    \qhww{2x^2+3x-5}{2x^2+5x-2x-5}
    \qhww{3x^2+2x-5}{3x^2+5x-3x-5}
    \qhww{3x^2+14x-5}{3x^2+15x-x-5}
    \qhww{2x^2-7x-15}{2x^2-10x+3x-15}
    \qhww{2x^2+x-15}{2x^2-6x+5x-15}
    \qhww{6x^2+17x-3}{6x^2+18x-x-3}
    \qhww{6x^2-7x-3}{6x^2-9x+2x-3}
    \qhww{6x^2+5x-6}{6x^2+9x-4x-6}
    \qhww{5x^2+23x+12}{5x^2+20x+3x+12}
    \qhww{5x^2+4x-12}{5x^2+10x-6x-12}
    \qhww{5x^2-19x+12}{5x^2-15x-4x+12}
    \qhww{5x^2-11x-12}{5x^2-15x+4x-12}
    \qhww{5x^2+28x-12}{5x^2+30x-2x-12}
    \qhww{9x^2-6x-8}{9x^2-12x+6x-8}
    \qhww{3x^2+13x-30}{3x^2-18x+5x-30}

\begin{singlespace}
Take a careful look at what you achieved in the previous question. An expression like
$3x^2+4x+1$ has no common factors whatsoever, but you managed to factorise it to
$(3x+1)(x+1)$. This was achieved by \emph{splitting the middle term} (the term in~$x$)
into two terms, then factorising by grouping in pairs.\par

In the next question, you will factorise more of these expressions but you will not be
given the first line of working. You will need to split the middle term yourself.
Hopefully, by closely observing all the examples in the previous question, you will be
able to work out how to do this. If not, help is available.
\end{singlespace}

  \question{5}{Factorise these quadratic trinomials:}
    \qwww{3x^2+8x+4}
    \qwww{5x^2+17x+6}
    \qwww{4x^2+13x+3}
    \qwww{5x^2+16x+11}
    \qwww{5k^2-12k+4}
    \qwww{9m^2-9m-4}
    \qwww{4x^2+4x-15}
    \qwww{3x^2-7x-6}
    \qwww{5x^2-14x+8}
    \qwww{2x^2-11x+12}
    \qwww{5x^2-16x+11}
    \qwww{8x^2-2x-3}
    \qwww{6x^2-x-7}
    \qwww{3x^2+2x-16}

\needspace{5\baselineskip}
\begin{singlespace}
The following questions have something in common.
\end{singlespace}

  \question{6}{Factorise:}
    \qwww{4x^2+12x+9}
    \qwww{9x^2-6x+1}
    \qwww{9x^2+6x+1}
    \qwww{16x^2+24x+9}

\begin{singlespace}
Finally, the next questions are \emph{monic} quadratics: their \emph{leading coefficient}
is $1$. Follow the same working as usual for all of these questions, but see if you can
find a shortcut to use in the next question.
\end{singlespace}

  \question{7}{Factorise, showing all working:}
    \qhww{m^2+7m+6}{m^2+6m+m+6}
    \qhww{a^2+12a+20}{a^2+10a+2a+20}
    \qwww{k^2+5k+6}
    \qwww{x^2-4x+3}
    \qwww{c^2-7c+10}
    \qwww{a^2-7a+12}
    \qwww{w^2-2w-8}
    \qwww{p^2-2p-15}
    \qwww{y^2+3y-28}
    \qwww{x^2+3x-40}
    \qwww{a^2+6a+9}
    \qwww{b^2-8b+16}

  \question{8}{Factorise, using a shortcut:}
    \qw{x^2+8x+15}
    \qw{y^2+9y+20}
    \qw{b^2-8b+12}
    \qw{t^2+t-2}
    \qw{u^2-u-2}
    \qw{a^2+2a-8}
    \qw{c^2-12c+27}
    \qw{u^2-13u+42}
    \qw{x^2-x-90}
    \qw{a^2-4a-32}
    \qw{y^2+23y-50}
    \qw{u^2-16u-80}

  \question{9}{Solve the following crpytic clues:}
  \begin{itemize}
    \item Artist changes channel (6)
    \item Billy's mate's a babysitter (5)
    \item Brutal pharaoh imprisoned Greek character (5)
    \item You can beat these urges (4)
    \item Yes, dad, news was broken in Midweek Times (10)
    \item Ripped tuxedo edges again (4)
    \item Inactive geek lost 500 in infotech (5)
    \item Lift in right aisle has top light out (5)
    \item Spooner cutting grass with eyes closed? Amazing! (4,7)
    \item Junks get zany on this river (7)
  \end{itemize}

\vspace{5mm}

The answers are, in order:
  \begin{itemize}
    \item \_ \_ R \_ \_ T
    \item N \_ \_ \_ \_
    \item \_ L \_ H \_
    \item E \_ \_ \_
    \item \_ E \_ N \_ \_ \_ A \_ \_
    \item \_ \_ R \_
    \item I \_ \_ \_ T
    \item R \_ I \_ \_
    \item M \_ N \_~~~B \_ \_ \_ \_ \_ \_
    \item \_ \_ \_ \_ \_ \_ \_ 
  \end{itemize}

\begin{singlespace}
  To work them out, remember the following. Each clue has a \emph{definition} at the
  beginning or the end. (Two clues---the second and fourth---have the definition at
  \emph{both} ends.) The rest of the clue is \emph{wordplay}, which is another way of
  getting the answer. Look for anagrams (clued by words such as changes, broken, junks),
  hidden words (``imprisoned''), abbreviations (for ``right'' and ``again''), synonyms,
  even Roman numerals! Also, words can get sliced and diced: ``tuxedo edges'' =
  \texttt{TO} and ``top light'' = \texttt{L}.  Good luck!
\end{singlespace}

\vfill

\begin{singlespace}
\textbf{Credits}\quad Some questions are taken from \emph{Maths in Focus} and \emph{Cambridge
Mathematics}, both at the Year 11 Advanced level. Cryptic clues are from \emph{The
Sydney Morning Herald}.
\end{singlespace}

\end{multicols*}
\end{document}