有理表达式的多项式部分的选择性着色

2024-5-25 • tag-icon

我正在教高中生如何通过取消因数来简化有理表达式。

取消的表达式应为红色（包括 L/R 括号），但分数线不为红色。下图显示了我目前能够实现的目标以及仍需应用红色的地方：

MWE 如下。非常感谢您的帮助（我的学生也一样）。

\documentclass[12pt]{exam}
        
\printanswers
% un-comment to print solutions.
\renewcommand{\solutiontitle}{}

\usepackage{multirow, tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\usepackage[table]{xcolor}
\usepackage{amsmath}
\usepackage{cancel}
\usepackage{framed}
\usepackage{multicol}
\usepackage{tasks}
\usepackage[a4paper,margin=0.5in,include head]{geometry}

\everymath{\displaystyle}
\setlength\parindent{1em}

\pagestyle{head}
\header{Algebra II Review Ch 3.2: Operations Rational Expressions and Equations: K E Y}
       {}
       {01/13-14/21} 

\newcommand{\pagetop}{%
\noindent 
  \fbox{\fbox{\parbox{\dimexpr\textwidth-4\fboxsep-4\fboxrule}{
    \textbf {Obj. 3.2.a: I can simplify factored rational expressions and find their restrictions.
    \bigskip
    \bigskipSimplify expression and state the excluded values (+1 pt numerator, +1 pt denominator, +1 pt restrictions.) each equation. Show all work/steps on this page.}
  }}}
  \bigskip
  \vspace{0.5mm}
}

\settasks{after-item-skip=1em,
          after-skip=2cm,
          label-width=2em,
          item-indent=3em,
          label=(\arabic*),
          column-sep=2em
          }
% ------------ DOCUMENT STARTS HERE---------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

%definition for bigskip = 1 line to replace all \bigskip
\def\bigskip{\vskip\bigskipamount}

\begin{tasks}
[style=enumerate](2)
% Prob #1
\task $\dfrac{10k^2+32k+24}{15k+18}$
\begin{solutionorbox}[5cm]
Factor 2 out of the numerator.\bigskip

$\dfrac{2(5k^2+16k+12)} {15k+18}$\bigskip

The 5 in front of $5k^2$ means this is a non-monic quadratic trinomial. So use the box method to complete the factorization.

\newcommand\mcc[1]{\multicolumn{1}{c}{#1}} 
\hspace{1cm}
\renewcommand\arraystretch{2}
\begin{tabular}{ c | c | c | }
  \mcc{} & \mcc{\textcolor{red}{$5k$}} & \mcc{\textcolor{red}{$+6$}} \\
    \cline{2-3}
 \textcolor{red}{$k$} & $5k^2$ & $6$ \\
    \cline{2-3}
 \textcolor{red}{$+2$} & $10k$ & $12$ \\
    \cline{2-3}
\end{tabular}

\vspace{0.2cm}
Numerator factors to $\color{red}\dfrac{(5k+6)(k+2)}{\color{black} 15k+18}$ 

Now factor the bottom:

\hspace{2cm}$3(5k+6)$
\vspace{0.25cm}

Cancel common factors (this creates a HOLE in the graph):

\hspace{2cm}
$\dfrac{\cancel{(5k+6)}(k+2)}{3\cancel{(5k+6)}}$ 
\vspace{0.25cm}

Simplified form: $\dfrac{(k+2)}{3}$

\textcolor{blue}{\textbf {Reminder:}}

\textbf{Zeros occur on top} (in the numerator).

\textbf{VA's (vertical asymptotes) are restrictions in the denominator}...to prevent division by $0$.

\textcolor{red}
{Zeros: $k=-2$}

\textcolor{red}
{Holes: $k=-6/5$}

\textcolor{red}
{VA: $none$}

\end{solutionorbox}

\vspace{0.25cm}
% Prob #1
\task $\dfrac{5k^2+10k+24}{6k+12}$
\begin{solutionorbox}[5cm]
step-by-step solution goes here
\end{solutionorbox}
\end{tasks}
\end{document}

答案1

\textcolor{red}{<stuff>}如果没有其他可用的范围，则主要需要用红色突出显示各个组件：

\documentclass{article}

\usepackage[table]{xcolor}
\usepackage{amsmath}
\usepackage{cancel}

\begin{document}

Numerator factors to
\[
  \dfrac{\color{red}(5k + 6)(k + 2)}{15k + 18}
\]
Cancel common factors (this creates a HOLE in the graph):
\[
  \dfrac{\textcolor{red}{\cancel{(5k + 6)}}(k + 2)}{3 \textcolor{red}{\cancel{(5k + 6)}}} 
\]

\end{document}

答案1

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