Multivariate polynomial long division using LaTeX
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
.
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:
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{22}
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 thecalc
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}
\[
\begin{array}{rr}
a_1\colon & \CenterInCol{y}\\
a_2\colon & \CenterInCol{1}\\
xy + 1\PhantC & \multirow{2}*{$\sqrt{xy^2 + 1}$}\\
y + 1\PhantC &\\
& xy^2 + y\\\cline{22}
& y + 1 \\
& y  1 \\\cline{22}
& 2
\end{array}
\]
\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/wrappingcmidruleinamacro
\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}
\[
\begin{array}{rll}
a_1\colon & \multicolumn{1}{l}{x+y}\\
a_2\colon & \multicolumn{1}{l}{1} & r\\
xy + 1\PhantC & \multirow{2}*{$\sqrt{x^2y + xy^2+y^2}$} \\\cline{33}
y^2 + 1\PhantC &\\
%
&\PhantSQ x^2y  x \\\CMidRule[3.0ex][9.0ex]{22}
&\PhantSQ \hphantom{x^2y +{}} xy^2 + x + y^2 \\
&\PhantSQ \hphantom{x^2y +{}} xy^2  y \\\CMidRule[9.0ex][5.0ex]{22}
&\PhantSQ \hphantom{x^2y + xy^2 +{}} x + y^2 + y \\\CMidRule[16.0ex][5.0ex]{22}
&\PhantSQ \hphantom{x^2y + xy^2 + x +{}} y^2 + y \\
&\PhantSQ \hphantom{x^2y + xy^2 + x +{}} y^2  1 \\\CMidRule[20.0ex][5.0ex]{22}
&\PhantSQ \hphantom{x^2y + xy^2 + x + y^2 +{}} y + 1 \\\CMidRule[25.0ex][1.0ex]{22}
&\PhantSQ \hphantom{x^2y + xy^2 + x + y^2 + y +{}} 1 &\to x+y \\\CMidRule[25.0ex][1.0ex]{22}
&\PhantSQ \hphantom{x^2y + xy^2 + x + y^2 + y +{}} 0 &\to x+y+1
\\&\PhantSQ\color{red}x^2y + xy^2 + x + y^2 + y + 1
% Last line above should be removed  used for alignment purposes only.
\end{array}
\]
\end{document}
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Katherine M.
Updated on June 12, 2020Comments

Katherine M. over 3 years
How do I typeset polynomial long division using multiple variables in LaTeX? I am hoping to be able to do polynomial long division using 2 or more divisors. Currently, the packages I am using are:
\usepackage[psamsfonts]{amsfonts} \usepackage{sudoku} \usepackage{natbib} \usepackage{amsmath, amsthm, amssymb} \usepackage{epsfig} \usepackage{makeidx} \usepackage{array} \usepackage{enumerate} \usepackage[usenames]{color} \usepackage{setspace} \usepackage{latexsym} \usepackage[dvips]{graphicx} \usepackage{eucal} \usepackage{polynom}
I know that there are a lot of packages that I am using which are unrelated to my question, but I wouldn't want to use conflicting packages.
I thought the the
polynom
package would be my solution, but one of the disclaimers in the package is that "Multivariate polynomials are currently not supported."Any help would be greatly appreciated.

Peter Grill over 11 yearsI am not understanding this. Can you show an image of what the desired output is preferably with a MWE that shows what you have attempted and the specific problem you have encountered.

Katherine M. over 11 yearsI have no idea how to even come close to writing the code that I need for this. The best that I can do is reference a paper that uses what I'm looking for: math.berkeley.edu/~aboocher/kaitlyn.pdf The fourth page has the type of division that I'm trying to do.

Peter Grill over 11 yearsCan you be a bit more specific as to exactly where on the paper. I appreciate that you can not produce this, but it is usually a good idea to show what you tried.

Katherine M. over 11 yearsIn the middle of the fourth page and on the top of the fifth page, she does polynomial long division with multivariables. I have tried using the polynom package but I am not at all familiar with code and it doesn't allow what I need. What I am trying now is using excel to write the formulas, saving it as a pdf, and inserting that into LaTeX. However, that looks extremely messy and unprofessional.


Katherine M. over 11 yearsThank you very much for your help! Is there a way to change the alignment of the terms being subtracted? If you look at the top of pg 5 of Kaitlyn Moran's document, math.berkeley.edu/~aboocher/kaitlyn.pdf she has multiple terms and I was hoping to have a similar effect.

Peter Grill over 11 years@KatherineM.: If there are not a large number of these I would resort to using
\hphantom{}
to leave the exact amount of space. Otherwise you could use multiple align points within the array. 
Katherine M. over 11 yearsDo you have any suggestions for helpful websites to help me better understand how arrays work?

Peter Grill over 11 yearsThere should be numerous examples on this site, very similar to
tabular
except in math mode. I will try to do a solution that uses\phantom{}
to do the spacing. Better still is if you attempt to produce your version, and then you could post a question here as to how to do better alignment. 
Peter Grill over 11 years@KatherineM.: Updated solution. Let me know if something is not clear.