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LaTeX 中多项式除法的自动化盒子方法/面积模型

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LaTeX 中多项式除法的自动化盒子方法/面积模型

我正在使用面积模型（通常称为“盒子方法”）编写多项式除法的工作表：

在此处输入图片描述

对于数十个问题，这个过程非常缓慢，尤其是当问题较大时，例如用五次多项式除以三次多项式。将来有没有办法自动创建这些框划分？

我希望能够输入类似的内容\polyboxdiv{6x^3+25x^2-24x+11}{x+5}（类似于\polylongdiv{6x^3+25x^2-24x+11}{x+5}来自polynom包的内容）来自动生成框划分表示。

这里有一个关联致 MWE：

\documentclass[addpoints]{exam}
\usepackage[utf8]{inputenc}
\usepackage[margin=.75 in]{geometry}
\usepackage{amsmath,amsfonts,amssymb,amsthm,color,srcltx,enumitem,bm,cancel,thmtools,physics}
\usepackage{multicol} %see http://stackoverflow.com/questions/1398127/breaking-a-list-into-multiple-columns-in-latex
\usepackage{multirow,array} %Used for the "hand-made" payoff matrix
\usepackage{hhline}
\usepackage{float}
\usepackage{polynom} %for polynomial long division and synthetic division, see http://texdoc.net/texmf-dist/doc/latex/polynom/polynom.pdf
\usepackage[table]{xcolor}

%For solution tables, see https://tex.stackexchange.com/questions/133397/printanswer-in-table-in-exam-class
\makeatletter
\newcommand{\st}[1]{\ifprintanswers\begingroup\Solution@Emphasis#1\if@shadedsolutions%
{\cellcolor{white}}
\else
\fi\endgroup\else\phantom{#1}\fi}
\makeatother
\SolutionEmphasis{\color{black}} %color font solutions
%\unframedsolutions  %Not necessary for table environment
%\shadedsolutions    %gives background color to solutions
\printanswers

\begin{document}
\section{Polynomials Division Test}
\begin{questions}

\question[40]
For the division problem
\begin{equation*}
(6x^3+25x^2-24x+11) \divisionsymbol (x+5)
\end{equation*}
\begin{parts}

\begin{multicols}{2}
\part Divide using an area model with a left divisor.
{\renewcommand{\arraystretch}{2}
\begin{table}[H]
\centering
\begin{tabular}{*{1}{>{\centering\arraybackslash}m{1.25cm}}|*{4}{>{\centering\arraybackslash}m{1.25cm}|}}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{$\st{6x^2}$}  & \multicolumn{1}{c}{$\st{-5x}$} & \multicolumn{1}{c}{$\st{+1}$} & \multicolumn{1}{c}{} \\ \hhline{~----}
\multirow{2}*{} $\st{x}$   & $\st{6x^3}$    & $\st{-5x^2}$   & $\st{x}$ & $\st{6}$ \\ \hhline{~----}
                $\st{+5}$  & $\st{30x^2}$   & $\st{-25x}$    & $\st{5}$ & \cellcolor{black!10} \\ \hhline{~----}
\end{tabular}
\end{table}}
\columnbreak
\part Divide using an area model with an upper divisor.
{\renewcommand{\arraystretch}{2}
\begin{table}[H]
\centering
\begin{tabular}{*{1}{>{\centering\arraybackslash}m{1.25cm}}|*{2}{>{\centering\arraybackslash}m{1.25cm}|}}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{$\st{x}$}  & \multicolumn{1}{c}{$\st{+5}$} \\ \hhline{~--}
\multirow{2}*{} $\st{6x^2}$ & $\st{6x^3}$   & $\st{30x^2}$          \\ \hhline{~--}
                $\st{-5x}$  & $\st{-5x^2}$  & $\st{-25x}$           \\ \hhline{~--}
                $\st{+1}$   & $\st{x}$      & $\st{5}$              \\ \hhline{~--}
                            & $\st{6}$      & \cellcolor{black!10}  \\ \hhline{~--}
\end{tabular}
\end{table}}
\end{multicols}
\vspace*{\stretch{1}}

\begin{multicols}{2}
\part Divide using the classic long-division algorithm.
\begin{solution}
\begin{center}
\polylongdiv{6x^3+25x^2-24x+11}{x+5}
\end{center}
\end{solution}
\columnbreak
\part Divide using Horner's method of synthetic division.
\begin{solution}
\begin{center}
\polyhornerscheme[x=-5]{6x^3+25x^2-24x+11}
\end{center}
\end{solution}
\end{multicols}
\vspace*{\stretch{1}}

\end{parts}
\end{questions}
\end{document}

谢谢你！

答案1

我可以帮你解答部分问题：如何自动化解决问题。但是，格式化相当繁琐。顶行是商，最右边一列的框是余数。计算机代数系统智者基于 Python 的 可以轻松告诉我们 的商和余数q,r = numerator.quo_rem(denominator)。 函数FormatTerm(a,deg)将帮助格式化各个项，函数PolyBoxDivL(f,g)将格式化表格。不过，效果不如你想要的那么好。

\documentclass[addpoints]{exam}
\usepackage[utf8]{inputenc}
\usepackage[margin=.75 in]{geometry}
\usepackage{amsmath,amsfonts,amssymb,amsthm,color,srcltx,enumitem,bm,cancel,thmtools,physics}
\usepackage{multicol}
\usepackage{multirow,array}
\usepackage[table]{xcolor}
\usepackage{sagetex}          %gives us access to SAGE
%%%%%%%%%%%%%%%
\begin{document}
\section{Polynomials Division Test}
\begin{sagesilent}
R.<x> = PolynomialRing(ZZ)    #### Ring of polynomials with integer coefficients
def FormatTerm(a,deg):
    if deg == 0:
        return "$%s$"%(a)

    if deg == 1:
        if a == 1:
            return "$x$"
        elif a == -1:
            return "$-x$"
        else:
            return "$%s x$"%(a)

    if deg >1:
        if a == 1:
            return "$x^{%s}$"%(deg)
        else:
            return "$%s x^{%s}$"%(a,deg)

def PolyBoxDivL(f,g):
    numerator = f
    denominator = g
    q,r = numerator.quo_rem(denominator)
    degreeq = q.degree()
    length = q.number_of_terms()+2   ##### +1 to include the remainder and vertical poly
    width = g.number_of_terms()
########### Table header    
    output = r""
    output += r"\begin{tabular}{c|"+"c|"*(length)+"}"
    output += r"\cline{2-%s}"%(length)
    output += r" & "
    for i in range(degreeq,-1,-1):
        if q[i] != 0:
            output += r"%s & "%(FormatTerm(q[i],i))
    output += r"\rule{0pt}{13pt} \\[6pt]\cline{2-%s}"%(length)
#### rows which aren't the header 

    for i in range(g.degree(),-1,-1):
        if denominator[i] != 0:
            output += r"\rule{0pt}{18pt} %s &"%(FormatTerm(denominator[i],i))
            for j in range(q.degree(),-1,-1):
                if q[j] != 0:
                    output += r"%s & "%(FormatTerm(denominator[i]*q[j],i+j))
            if i == g.degree():
                output += r"$%s$ \\[8pt]\cline{2-%s}"%(latex(r),length)
            else:
                output += r"\cellcolor{black!10} \\[8pt]\cline{2-%s}"%(length)
    output += r"\end{tabular}"

    return output
\end{sagesilent}

Use the area model with a left divisor to find $(6x^3+25x^2-24x+11)   \divisionsymbol (x+5)$:\\\\
\begin{sagesilent}
Q1 = PolyBoxDivL(6*x^3+25*x^2-24*x+11,x+5)
\end{sagesilent}
\sagestr{Q1}

\vspace{.5cm}
Use the area model with a left divisor to find $(5x^5-3x^2+x-1) \divisionsymbol (x^2+x+1)$:\\\\
\begin{sagesilent}
Q2 = PolyBoxDivL(5*x^5-3*x^2+x-1,x^2+x+1)
\end{sagesilent}
\sagestr{Q2}

\vspace{.5cm}
Use the area model with a left divisor to find $(x^4-1) \divisionsymbol (x^2-1)$:\\\\
\begin{sagesilent}
Q3 = PolyBoxDivL(x^4-1,x^2-1)
\end{sagesilent}
\sagestr{Q3}

\vspace{.5cm}
Use the area model with a left divisor to find $(x^9-7x^4) \divisionsymbol (x^3-x+4)$:\\\\
\begin{sagesilent}
Q4 = PolyBoxDivL(x^9-7*x^4,x^3-x+4)
\end{sagesilent}
\sagestr{Q4}
\end{document}

SAGE 不是 LaTeX 的一部分。它需要下载到您的机器上，或者更简单的是，通过免费的可钙账户。Cocalc 中运行的代码如下所示：

在此处输入图片描述

需要注意的非常重要的一行是output += r"\begin{tabular}{c|"+"c|"*(length)+"}"。这允许表格改变列数，因为 SAGE 可以计算计算列数所需的非零项数。具有上除数（？）的面积模型以类似的方式处理。问题通过 计算，\begin{sagesilent} Q4 = PolyBoxDivL(x^9-7*x^4,x^3-x+4) \end{sagesilent}然后将结果字符串放入带有 的文档中\sagestr{Q4}。

编辑：为了回应下面的评论，我添加了 0 系数项，以便指数在对角线上相同。我不知道如何让表格格式处理某些行的垂直线。我选择给单元格上色来解决这个问题

\documentclass[addpoints]{exam}
\usepackage[utf8]{inputenc}
\usepackage[margin=.75 in]{geometry}
     \usepackage{amsmath,amsfonts,amssymb,amsthm,color,srcltx,enumitem,bm,cancel,thmtools,physics}
\usepackage{multicol}
\usepackage{multirow,array}
\usepackage[table]{xcolor}
\usepackage{sagetex}          %gives us access to SAGE
%%%%%%%%%%%%%%%
\begin{document}
\section{Polynomials Division Test}
\begin{sagesilent}
R.<x> = PolynomialRing(ZZ)    #### Ring of polynomials with integer   coefficients
def FormatTerm(a,deg):
    if deg == 0:
        return "$%s$"%(a)

    if deg == 1:
        if a == 1:
            return "$x$"
        elif a == -1:
            return "$-x$"
        else:
            return "$%s x$"%(a)

    if deg >1:
        if a == 1:
            return "$x^{%s}$"%(deg)
        else:
            return "$%s x^{%s}$"%(a,deg)

def PolyBoxDivL(f,g):
    numerator = f
    denominator = g
    q,r = numerator.quo_rem(denominator)
    length = q.degree()+3
    width = g.degree()+1
########### Table header    
    output = r""
    output += r"\begin{tabular}{"+"c"*(length)+"}"
    output += r" & "
    for i in range(q.degree(),-1,-1):
        output += r" \cellcolor{black!10} %s & "%(FormatTerm(q[i],i))
    output += r"\rule{0pt}{13pt} \\[6pt]"
#### rows which aren't the header

    for i in range(g.degree(),-1,-1):
        output += r"\rule{0pt}{18pt} \cellcolor{black!10} %s &"%(FormatTerm(denominator[i],i))
        for j in range(q.degree(),-1,-1):
            output += r"\cellcolor{orange!10} %s & "%(FormatTerm(denominator[i]*q[j],i+j))
        if i == g.degree():
            output += r"\cellcolor{blue!10} $%s$ \\[8pt]"%(latex(r))
        else:
            output += r"\cellcolor{blue!10} \\[8pt]"
    output += r"\end{tabular}"

    return output
\end{sagesilent}

Use the area model with a left divisor to find $(6x^3+25x^2-24x+11)     \divisionsymbol (x+5)$:\\\\
\begin{sagesilent}
Q1 = PolyBoxDivL(6*x^3+25*x^2-24*x+11,x+5)
\end{sagesilent}
\sagestr{Q1}

\vspace{.5cm}
Use the area model with a left divisor to find $(5x^5-3x^2+x-1) \divisionsymbol (x^2+x+1)$:\\\\
\begin{sagesilent}
Q2 = PolyBoxDivL(5*x^5-3*x^2+x-1,x^2+x+1)
\end{sagesilent}
\sagestr{Q2}

\vspace{.5cm}
Use the area model with a left divisor to find $(x^4-1) \divisionsymbol (x^2-1)$:\\\\
\begin{sagesilent}
Q3 = PolyBoxDivL(x^4-1,x^2-1)
\end{sagesilent}
\sagestr{Q3}

\vspace{.5cm}
Use the area model with a left divisor to find $(x^9-7x^4) \divisionsymbol (x^3-x+4)$:\\\\
\begin{sagesilent}
Q4 = PolyBoxDivL(x^9-7*x^4,x^3-x+4)
\end{sagesilent}
\sagestr{Q4}
\end{document}

输出为

在此处输入图片描述

答案2

我所描述的不是一个答案，而是一个为你指明正确方向的东西

由于它不适合放在评论栏，所以我把它发布在这里

希望版主不会介意！

你要问的是

  • 将多项式的值解析到表格环境中

  • 桌子设计有四种不同的设计

  • 从你在分子和分母部分输入多项式开始，这个过程就是自动化的

请参阅此网站上的答案

使用文本中的引用自动从数​​据工具创建表格

使用数据工具格式化 CSV 中的复杂表格

\begin{filecontents*}{\jobname.dat}
  Hammer001,   Hammer,    1 ,  0 , 1 , 10 , 1 , light (add some words here to wrap around)
  Hammer002,   Hammer,    2 ,  0 , 1 , 10 , 1 , heavy
  Hammer003,   Hammer,    3 ,  0 , 1 , 10 , 1 , really heavy
  Longsword001,Longsword, 1 , -1 , 2 , 75 , 2 , one-handed
  Longsword002,Longsword, 2 , -1 , 2 , 75 , 2 , two-handed
  Longsword003,Longsword, 3 , -1 , 2 , 75 , 2 , three-handed
\end{filecontents*}

\documentclass{article}
\usepackage{array}
\usepackage{datatool}
\usepackage{longtable}
\usepackage{etoolbox}

\newcommand{\colhead}[1]{\multicolumn{1}{>{\bfseries}l}{#1}}
\newcounter{tabenum}\setcounter{tabenum}{0}
\newcommand{\nextnuml}[1]{%
  \refstepcounter{tabenum}\thetabenum.\label{#1}%
}

\newcommand{\PrintDTLTable}[2][]{%
 % #1 = list of rowIDs
 % #2 = database to search
  \begin{longtable}{l l l l l l l m{2in}}
  & \colhead{Label} & \colhead{Cost} & \colhead{Weight} &
    \colhead{PropA} & \colhead{PropB} & \colhead{PropC} & \colhead{Description}\\
  \hline
  \DTLforeach
    [%
     \(\equal{#1}{}\AND\DTLisSubString{\ReferencedIDs}{\RowID}\)
     \OR
     \(\DTLisSubString{#1}{\RowID}\AND\DTLisSubString{\ReferencedIDs}{\RowID}\)%
    ]
    {#2}{%
      \RowID=RowID,%
      \Label=Label,%
      \Cost=Cost,%
      \Weight=Weight,%
      \PropA=PropA,%
      \PropB=PropB,%
      \PropC=PropC,%
      \Description=Description%
     }{%
       \nextnuml{\RowID} & \Label &\Cost & \Weight & \PropA & \PropB & \PropC & \Description \\
    }%
    \end{longtable}
}

\makeatletter
\let\oldref\ref
\def\ref#1{%
  \immediate\write\@auxout{%
    \string\gappto\string\ReferencedIDs{#1,}%
  }%
  \oldref{#1}%
}
\def\ReferencedIDs{}
\makeatother


\begin{document}

% \DTLsetseparator{&}% Define separator of the data
\DTLloaddb[noheader,keys={RowID,Label,Cost,Weight,PropA,PropB,PropC,Description}]{myDB}{\jobname.dat}

% \DTLdisplaydb{myDB}% Useful for debugging.

\PrintDTLTable[Hammer001,Hammer003,Longsword003]{myDB}

\PrintDTLTable[Hammer002,Longsword002]{myDB}

This is a reference to ~\ref{Hammer003}.

This is a reference to ~\ref{Longsword002}.

\end{document}

表格形式的输出如下

在此处输入图片描述

第一部分代码

\begin{filecontents*}{\jobname.dat} Hammer001, Hammer, 1 , 0 , 1 , 10 , 1 , light (add some words here to wrap around) Hammer002, Hammer, 2 , 0 , 1 , 10 , 1 , heavy Hammer003, Hammer, 3 , 0 , 1 , 10 , 1 , really heavy Longsword001,Longsword, 1 , -1 , 2 , 75 , 2 , one-handed Longsword002,Longsword, 2 , -1 , 2 , 75 , 2 , two-handed Longsword003,Longsword, 3 , -1 , 2 , 75 , 2 , three-handed \end{filecontents*}

类似于对多项式进行广告宣传，然后将其解析为表格形式

网页上有类似的查询，您可能想探索一下

在此处输入图片描述

我相信本网站的一位专家很快就会回复您

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