12

Original

The remainder provided by polynom.sty is NOT complete (as shown in the following figure).

alt text


My Objective

When I was in senior high school, my teacher made the diagram like below. :-)

Thus the final step must show the TOTAL remainder and mark it with L-shaped curve.

NOTE

I am sorry. The previous post I forgot to put back -1/4 in the original position. It have made you confused. Now there is no mistake anymore. :-)

alt text


Minimal Code

\documentclass{article}
\usepackage{polynom,array}
\usepackage[table]{xcolor}


\makeatletter
\def\pld@ArrangeResult#1{%
    \ifx\pld@remainder\@empty
        \@tempcnta\pld@maxcol\relax
        \pld@InsertItems@do\pld@lastline
            {\pld@firsttrue\pld@PLD{\pld@R{0}{1}}}%
    \fi
    \ifnum\pld@currstage>\z@
        \pld@Extend\pld@allines{\pld@lastline\cr}%
    \else
        \pld@InsertFake\pld@lastline
    \fi
    \pld@iftopresult
        \def\pld@lastline{\pld@PrintPoly\pld@divisor%
        %====================================================================================
        \quad\smash{{\color{red}\rule[-6pt]{\arrayrulewidth}{17pt}}}\kern-\arrayrulewidth&}%
        %====================================================================================
    \else
        \let\pld@lastline\@empty
        \ifx B\pld@style\else
            \def\pld@lastline{\pld@leftdelim\strut\pld@rightxdelim&}%
        \fi
    \fi
    \expandafter\pld@AR@col\expandafter\pld@PLD
                           \expandafter\pld@lastline#1+\relax+%
    \pld@SplitQuotient
    \pld@iftopresult
        \let\pld@currentline\@empty
        \expandafter\pld@AR@col\expandafter\pld@PLD
                               \expandafter\pld@currentline
                                           \pld@quotient+\relax+%
        \expandafter\pld@AR@col\expandafter\pld@XPLD
                               \expandafter\pld@currentline
                                           \pld@shadow+\relax+%
        \edef\pld@subline{%
            \noexpand\cline{\tw@-\pld@maxcol}%
            \noalign{\vskip\jot}}%
        \pld@Extend\pld@currentline{\expandafter\cr\pld@subline}%
    \else
        \@tempcnta-\@tempcnta
        \advance\@tempcnta\pld@maxcol\relax \advance\@tempcnta\@ne
        \edef\pld@span{\the\@tempcnta}%
        \ifx B\pld@style
          \pld@AddTo\pld@lastline{%
            &\multispan\pld@span${}=%
            \pld@PrintPolyWithDelims\pld@divisor
            \expandafter\pld@IfSum\expandafter{\pld@divisor}{}{\cdot}%
            \expandafter\pld@IfSum\expandafter{\pld@quotient}\pld@true
                                                             \pld@false
            \pld@if \pld@leftdelim
                    \pld@PrintPolyShadow
                    \pld@rightdelim
              \else \pld@PrintPolyShadow \fi
            \pld@firstfalse
            \expandafter\pld@PrintRemain\expandafter{\pld@remainder}$}%
        \else
          \pld@AddTo\pld@lastline{%
            &\multispan\pld@span$\pld@leftxdelim\strut\pld@rightdelim
            \pld@div
            \pld@PrintPolyWithDelims\pld@divisor=
            \pld@PrintPolyShadow
            \ifx\pld@remainder\@empty\else
                +{}%
                \setbox\z@=\hbox{$\displaystyle
                  \frac{\let\strut\@empty\pld@firsttrue \expandafter
                        \pld@PrintRemain\expandafter{\pld@remainder}}%
                       {\let\strut\@empty\pld@PrintPoly\pld@divisor}$}%
                \dp\z@=\z@\box\z@
            \fi
            $}%
        \fi
    \fi
\expandafter\pld@AR@\pld@allines\relax}
\makeatother


\arrayrulecolor{red}
\arrayrulewidth=0.8pt
\def\strut{\rule[-6pt]{0pt}{12pt}}

\begin{document}
\polylongdiv[style=A]{\frac{3}{7}x^9+x^2-\frac{1}{4}}{\frac{9}{5}x^4-1}

\vspace{1cm}
\polylongdiv[style=A]{x^4-1}{x^2-1}
\end{document}

4
  • @Herbert + @Ulrike: Please see my updated figure (second figure). When I was in senior high school, my teacher made the diagram like that. :-) He stopped the operation until showing the total remainder. Jan 12, 2011 at 10:12
  • I guess you don't want to move the 1/4 down, but you want to copy it. After all, you want to divide \frac{3}{7}x^9+x^2-\frac{1}{4} (including the 1/4) by \frac{9}{5}x^4-1. Jan 12, 2011 at 12:18
  • I've told you before that it's not necessary to write the whole question in the title of the question :-) Jan 12, 2011 at 13:41
  • @Hendrik: You are correct. I have to say "copy" instead of "drop". And I did NOT realize that delete -1/4 from the original position. That was my big mistake and my teacher did NOT teach like that. I am thinking to shrink my title. Thanks. Jan 12, 2011 at 13:48

2 Answers 2

10
+500

The following code should work; I've included a few test cases. To keep the code shorter, I didn't include your changes to the style of the output (red color, and bar between dividend and divisor instead of parenthesis). My changes and additions to polynom are marked with !!!.

\documentclass{article}
\usepackage{polynom,array}

\makeatletter
\def\pld@DivPoly@l{%
    \ifx\pld@remainder\@empty\else
        \pld@IfNeedsDivision\pld@remainder\pld@divisor
        {\pld@ExtendPoly\pld@quotient\pld@factor
         \pld@NMultiplyPoly\pld@sub\pld@divisor\pld@factor
         \pld@SubtractPoly\pld@remainder\pld@sub
         \expandafter\pld@DivPoly@l}%
        {\expandafter\pld@insert@remainder                       % !!!
         \pld@last@remainder+\relax\relax}                       % !!!
    \fi}
\def\pld@insert@remainder#1+#2\relax{%                           % !!!
    \ifx\relax#1\relax\else\pld@InsertItems\@empty\@empty{#1}\fi % !!!
    \ifx\relax#2\relax\else\pld@insert@remainder#2\relax\fi}     % !!!
\def\pld@SubtractPoly@l#1+#2\@empty#3+#4\@empty{%
    \ifx\relax#1\relax
        \let\pld@last@remainder\@empty                           % !!!
        \ifx\relax#3\relax \let\pld@next\@empty \else
          \pld@AddToPoly\pld@tempoly{#3}%
          \pld@if \pld@InsertItems{#3}{#3}{}\fi
          \def\pld@next{\pld@SubtractPoly@l\relax+\@empty#4\@empty}%
        \fi
    \else
    \ifx\relax#3\relax
        \pld@SubtractPoly@r#1+#2\@empty
        \let\pld@next\@empty
    \else
        \pld@IfMonomE{#1}{#3}%
        {\def\pld@temp{#1+#3}%
         \pld@CondenseMonomials\pld@true\pld@temp
         \ifx\pld@temp\@empty\else
             \pld@ExtendPoly\pld@tempoly\pld@temp
         \fi
         \pld@if \expandafter\pld@InsertItems\expandafter
                 {\pld@temp}{#3}{#1}\fi
         \def\pld@next{\pld@SubtractPoly@l#2\@empty#4\@empty}}%
        {\pld@IfMonomL{#1}{#3}%
         {\pld@AddToPoly\pld@tempoly{#3}%
          \pld@if \pld@InsertItems{#3}{#3}{}\fi
          \def\pld@next{\pld@SubtractPoly@l#1+#2\@empty#4\@empty}}%
         {\pld@AddToPoly\pld@tempoly{#1}%
          \pld@if \pld@InsertItems{#1}{}{#1}\fi
          \def\pld@next{\pld@SubtractPoly@l#2\@empty#3+#4\@empty}}%
        }%
    \fi \fi
    \pld@next}
\def\pld@SubtractPoly@r#1+\relax+\@empty{%
    \pld@AddToPoly\pld@tempoly{#1}%
    \def\pld@last@remainder{#1}}                                 % !!!
\makeatother
\def\strut{\rule[-6pt]{0pt}{12pt}}

\begin{document}
\polylongdiv{x^5-1}{x-1}

\polylongdiv{x^5-x^2}{x^2-1}

\polylongdiv{\frac{3}{7}x^9+x^2-\frac{1}{4}}{\frac{9}{5}x^4-1}

\polylongdiv{x^9+x^2-1}{x^4-x}

\polylongdiv{x^{15}+1}{x^5+x^3+x+1}

\polylongdiv{x^4-1}{x^2-1}
\end{document}
2
  • Well done!!
    – user2478
    Jan 12, 2011 at 21:36
  • Thanks for this answer. I am really satisfied with it. The bounty will be given several hours later because the time barrier must be elapsed for 24 hours since the bounty is initiated. Jan 13, 2011 at 1:54
3

Well the 1/4 is not the remainder, it is the last part of the first argument. The line you are trying to change is the line which shows both polynomials. You can see this if you use a bit simpler arguments:

\documentclass{article}
\usepackage{polynom,array}
\usepackage[table]{xcolor}

\arrayrulecolor{red}
\arrayrulewidth=0.8pt
\def\strut{\rule[-6pt]{0pt}{12pt}}

\begin{document}
\polylongdiv[style=A]{x^3-1}{x^2+1}

\bigskip

\polylongdiv[style=B]{x^3-1}{x^2+1}

\bigskip

\polylongdiv[style=C]{x^3-1}{x^2+1}

\end{document}

Edit: The good news are that the remainder is accessible:

\documentclass{article}
\usepackage{polynom}
\usepackage[table]{xcolor}

\arrayrulecolor{red}
\arrayrulewidth=0.8pt
\def\strut{\rule[-6pt]{0pt}{12pt}}

\makeatletter
\renewcommand*\polylongdiv[1][]{%
    \begingroup
    \let\pld@stage\maxdimen \polyset{#1}%
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\pld@LongDividePoly\pld@polya\pld@polyb
                 \pld@PrintLongDiv\\                   
   $\expandafter\pld@PrintRemain\expandafter{\pld@remainder}$%new
    \endgroup \ignorespaces}}
\begin{document}
\polylongdiv[style=A,stage=5]{x^3-1}{x^2+1}

\bigskip

\polylongdiv[style=A]{\frac{3}{7}x^9+x^2-\frac{1}{4}}{\frac{9}{5}x^4-1}
\end{document}

But I don't have the time to find out how to insert it correctly in the array.

9
  • @Ulrike: Using your example, the remainder is -x-1. But style=A only produces the remainder = -x. Jan 12, 2011 at 9:41
  • @Ulrike: -1/4 in my example is the last term of the remainder. :-) Jan 12, 2011 at 9:46
  • @Herbert: In mathematics, the remainder of x^3-1 divided by x^2+1 is -x-1. The polynom should drop -1 to make a final remainder. Jan 12, 2011 at 10:04
  • @Herbert: My final remainder is not -1/4 only. But -1/4 is the last term of the remainder. If you see the updated figure (second figure), it is the final diagram when doing this calculation. Jan 12, 2011 at 10:31
  • @Herbert + @Ulrike: The polynom.sty does NOT do wrong calculation but I need a modification such that at the final step the diagram show the total remainder as what many students used. The polynom.sty diagram looks unfamiliar for some students because it stops the operation without showing the final remainder. Jan 12, 2011 at 10:37

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