[Tex/LaTex] Polynomial division code

polynom

I would like to give me an example of how to use the polynom package. More specifically, I want to write down on my latex editor the following polynomial division (source):

enter image description here

Thanks in advance.

Best Answer

It looks like people did not sure whether this is a "do for me" question or you are asking anything else. Let me start to clarify something...

First you mentions polynom, which is good at creating long division

with relatively easy code

\documentclass{article}
\usepackage{polynom}
\begin{document}
    \polylongdiv{X^3-12X^2-42}{X-3}
\end{document}

But its output is relatively difficult to modify since it is hard-coded using tabular internally. You would need to rewrite the output scheme. If it is not the desired, the remaining problem is whether you prefer polynom's default output or the cross one you provide?

Second, Mathjax, though extremely convenient over HTML, uses its own engine. It supports small amount of packages by rewriting them case by case. (Which may save some resources since one does not need line-breaking/page-breaking/etc anymore.) Thus it is hard to create a consistent user experience between MathJax and polynom, a probably-will-never-support package. You might either

Concentrate on LaTeX and use htlatex to convert to HTML
Write Mathjax-friendly syntax in your LaTeX file
Work on both LaTeX and MathJax and adopt that they are not not going to be the same.

Basic Example:

Here is an example of one way to reproduce the Division Algorithm from Page 4 of Grobner Bases and their Applications by Kaitlyn Moran using an array.

enter image description here

Notes:

multirow was used to center the \sqrt{xy^2 + 1} term in between the two rows.
\multicolumn{1}{c}{<text>} was used to center content within a cell

Extended Example

For the more complicated case (from Page 5 of the above referenced paper), where there are multiple align points, I would recommend using an \hphantom{} to leave the appropriate amount of horizontal spacing. The simplest way to do that is to start at the bottom and type in the polynomial using the widest terms at each point. This is shown in red below:

enter image description here

From this widest polynomial, work upwards and place the terms you don't want printed in an \hphantom{} to get the appropriate spacing. In the code below I carefully aligned the code so that the \hpantom{} is easier to visualize.

The other complication with this is that the horizontal rules do not extend across the entire width of the column, so a simple \cline{2-2} won't work in this case. Luckily, you can use \cmidrule from the booktabs package which accepts an optional parameter that allows for trimming on the left and right of the \cmidrule. Since the syntax of this is not quite standard I wrapped that in the \CMidRule{<left trim>}{<right trim>}{<columns>} to which you can specify how much trimming to do on each side. So \CMidRule{0.0ex}{0.0ex}{<columns>} is equivalent to cmidrule{<columns>}.

Notes

The portion in red is only used to illustrate the alignment points and should be deleted.
For more details about the \phantom macro, see Indent as much as the width of a word for a basic example, or How to align across ordinary text; as in breaking matrices, sets of equations, tables for a more elaborate usage.
One thing to be careful about is that if a trailing binary operator (+, or -) is part of the \hphantom{}, then an additional {} needs to be inserted to ensure that it is treated as a binary operator. See Phantom width of binary operator.
The trim lengths specified in the \CMidRule can be computed based on the length of the various portions of the code using the calc package. However, in this case, using actual numbers to tweak the trims was more flexible and resulted in code that was easier to read.
It turns out that wrapping cmidrule in a macro is not straightforward. The earlier solution presented here seemed to work only because there is a single use of this macro per line, in this specific case. When an attempt to use this macro more than once per line, problems arose and discussed at Wrapping \cmidrule in a macro.

Code: Basic Example

\documentclass{article}
\usepackage{multirow}

\newcommand{\PhantC}{\phantom{\colon}}%
\newcommand{\CenterInCol}[1]{\multicolumn{1}{c}{#1}}%

\begin{document}
\[
\begin{array}{rr}
   a_1\colon  & \CenterInCol{y}\\
   a_2\colon  & \CenterInCol{-1}\\
xy + 1\PhantC & \multirow{2}*{$\sqrt{xy^2 + 1}$}\\
 y + 1\PhantC &\\
              & xy^2 + y\\\cline{2-2}
              & -y + 1 \\
              & -y - 1 \\\cline{2-2}
              & 2
\end{array}
\]
\end{document}

Code: Extended Example

\documentclass{article}
\usepackage{xcolor}
\usepackage{multirow}
\usepackage{booktabs}
\usepackage{xparse}
\usepackage{calc}

\newcommand{\PhantC}{\phantom{\colon}}%
\newcommand{\PhantSQ}{\phantom{\sqrt{\hspace{0.3ex}}}}%

% https://tex.stackexchange.com/questions/63355/wrapping-cmidrule-in-a-macro
\ExplSyntaxOn
\makeatletter
\newcommand{\CMidRule}{\noalign\bgroup\@CMidRule{}}
\NewDocumentCommand{\@CMidRule}{
    m % Material to reinsert before cmidrule.
    O{0.0ex} % #1 = left adjust
    O{0.0ex} % #1 = right adjust
    m  %       #3 = columns to span
}{
    \peek_meaning_remove_ignore_spaces:NTF \CMidRule
      { \@CMidRule { #1 \cmidrule[\cmidrulewidth](l{#2}r{#3}){#4} } }
      { \egroup #1 \cmidrule[\cmidrulewidth](l{#2}r{#3}){#4} }
}
\makeatother
\ExplSyntaxOff

\begin{document}
\[
\begin{array}{rll}
     a_1\colon  & \multicolumn{1}{l}{x+y}\\
     a_2\colon  & \multicolumn{1}{l}{1}           & r\\
  xy + 1\PhantC & \multirow{2}*{$\sqrt{x^2y + xy^2+y^2}$} \\\cline{3-3}
 y^2 + 1\PhantC &\\
    %
    &\PhantSQ           x^2y -     x                                         \\\CMidRule[3.0ex][9.0ex]{2-2}
    &\PhantSQ \hphantom{x^2y +{}}  xy^2 +    x + y^2                         \\
    &\PhantSQ \hphantom{x^2y +{}}  xy^2 -    y                               \\\CMidRule[9.0ex][5.0ex]{2-2}
    &\PhantSQ \hphantom{x^2y +     xy^2 +{}} x +    y^2 +     y              \\\CMidRule[16.0ex][5.0ex]{2-2}
    &\PhantSQ \hphantom{x^2y +     xy^2 +    x +{}} y^2 +     y              \\
    &\PhantSQ \hphantom{x^2y +     xy^2 +    x +{}} y^2 -     1              \\\CMidRule[20.0ex][5.0ex]{2-2}
    &\PhantSQ \hphantom{x^2y +     xy^2 +    x +    y^2 +{}}  y + 1          \\\CMidRule[25.0ex][1.0ex]{2-2}
    &\PhantSQ \hphantom{x^2y +     xy^2 +    x +    y^2 + y +{}}  1 &\to x+y \\\CMidRule[25.0ex][1.0ex]{2-2}
    &\PhantSQ \hphantom{x^2y +     xy^2 +    x +    y^2 + y +{}}  0 &\to x+y+1
  \\&\PhantSQ\color{red}x^2y +     xy^2 +    x +    y^2 + y + 1
   % Last line above should be removed -- used for alignment purposes only.
\end{array}
\]
\end{document}

[Tex/LaTex] Polynomial Long Division Using Polynom

Changing the internal macros that are responsible for omitting zero terms you get what you want (but you have to specify the zero term in the input):

\documentclass{article}
\usepackage{polynom}
\makeatletter
\def\pld@CF@loop#1+{%
    \ifx\relax#1\else
        \begingroup
          \pld@AccuSetX11%
          \def\pld@frac{{}{}}\let\pld@symbols\@empty\let\pld@vars\@empty
          \pld@false
          #1%
          \let\pld@temp\@empty
          \pld@AccuIfOne{}{\pld@AccuGet\pld@temp
                            \edef\pld@temp{\noexpand\pld@R\pld@temp}}%
           \pld@if \pld@Extend\pld@temp{\expandafter\pld@F\pld@frac}\fi
           \expandafter\pld@CF@loop@\pld@symbols\relax\@empty
           \expandafter\pld@CF@loop@\pld@vars\relax\@empty
           \ifx\@empty\pld@temp
               \def\pld@temp{\pld@R11}%
           \fi
          \global\let\@gtempa\pld@temp
        \endgroup
        \ifx\@empty\@gtempa\else
            \pld@ExtendPoly\pld@tempoly\@gtempa
        \fi
        \expandafter\pld@CF@loop
    \fi}
\def\pld@CMAddToTempoly{%
    \pld@AccuGet\pld@temp\edef\pld@temp{\noexpand\pld@R\pld@temp}%
    \pld@CondenseMonomials\pld@false\pld@symbols
    \ifx\pld@symbols\@empty \else
        \pld@ExtendPoly\pld@temp\pld@symbols
    \fi
    \ifx\pld@temp\@empty \else
        \pld@if
            \expandafter\pld@IfSum\expandafter{\pld@temp}%
                {\expandafter\def\expandafter\pld@temp\expandafter
                    {\expandafter\pld@F\expandafter{\pld@temp}{}}}%
                {}%
        \fi
        \pld@ExtendPoly\pld@tempoly\pld@temp
        \pld@Extend\pld@tempoly{\pld@monom}%
    \fi}
\makeatother
\begin{document}
\polylongdiv{X^3+X^2+0X-1}{X-1}
\end{document}

enter image description here

Best Answer

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[Tex/LaTex] Polynomial Long Division Using Polynom

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