How can I get the ordinary polynomial long division by using polydiv like this?

Question

I want to get the following polynomial long division like this:

Polynomial Long Division

But as you know, when I use command \polylongdiv (package polynom), I always get the following:

polydiv result

How can I get the result as in the first picture? At present, I have no idea to do so. Here is my tex file:

\documentclass{article}

\usepackage{polynom}

\begin{document}

$$\polylongdiv{x^3-12x^2-42}{x-3}$$

\end{document}

Question is not the same, but the code in How to correct excessive lines at the corner and bad line spaces in polynom.sty? replaces the ")" with a (better placed) vertical line. (For reference: the code comes from http://www.latex-community.org/forum/viewtopic.php?p=30051#p30051, and was written by Thorsten Donig.) — Torbjørn T., Oct 28 '12 at 13:59
@azhi: Would you be able to adequately verbalize the difference between the two outputs? For example, do you really want elements like 0x? The horizontal alignment seems to be a major difference. To what extent? Do you want things to still line up with other elements horizontally? — Werner, Nov 01 '12 at 06:21
I want the result of the command $\polylongdiv{x^3-12x^2-42}{x-3}$ to be as that of the first picture. Yes, I do really want the elements like $0X$. But what I want most is that the third line of the second picture to be $x^3-3x^2$, other than $-x^3+3x^2$, the fifth line to be $-9x^2+27x$, other than $9x^2-27x$, the seventh line to be $-27x+81$, other than $27x-81$. — azhi, Nov 01 '12 at 11:24
the way that \polylongdiv is rendering it is strictly correct, aligning terms of the same power. — Nicholas Hamilton, Dec 15 '12 at 16:46
Azhi, I understand your question. You wanted that third line to represent the result of multiplying the current term in the quotient by the divisor. You do NOT want to show the opposite of that product. Did you ever find a way to achieve your goal? — Randy Scott, Sep 25 '15 at 16:16

score 8 · Answer 1 · answered Dec 14 '16 at 22:07

A while back, I had a go at reimplementing polynomial long division because I wanted something similar.

Save this as polydiv.sty

\ProvidesPackage{polydiv}
\RequirePackage{xparse,expl3}
\ExplSyntaxOn

\bool_new:N \l__poly_zeros_bool
\bool_new:N \l__poly_first_bool
\bool_new:N \l__poly_trailing_bool
\bool_new:N \l__poly_ptrailing_bool
\bool_new:N \l__poly_stage_bool
\bool_set_true:N \l__poly_stage_bool
\tl_new:N \l__poly_var_tl
\tl_new:N \l__poly_sep_tl
\tl_new:N \l__poly_longdiv_sep_tl
\tl_new:N \l__poly_oline_tl
\tl_new:N \l__poly_uline_tl
\tl_set:Nn \l__poly_var_tl {x}
\tl_set:Nn \l__poly_sep_tl {}
\tl_set:Nn \l__poly_longdiv_sep_tl {}
\int_new:N \l__poly_deg_int
\int_new:N \l__poly_pad_int
\int_new:N \l__poly_tmpa_int
\int_new:N \l__poly_tmpb_int
\int_new:N \l__poly_trailing_int
\int_new:N \l__poly_stage_int
\int_new:N \l__poly_cstage_int
\int_set:Nn \l__poly_pad_int{3}
\seq_new:N \l__poly_gtmpa_seq

\keys_define:nn { polynomial }
{
  stage .code:n = {
    \bool_set_false:N \l__poly_stage_bool
    \int_set:Nn \l__poly_stage_int {#1}
  },
  zeros .bool_set:N = \l__poly_zeros_bool,
  separator .tl_set:N = \l__poly_sep_tl,
  variable .tl_set:N = \l__poly_var_tl,
  var .tl_set:N = \l__poly_var_tl,
%  trailing .bool_set:N = \l__poly_trailing_bool
}

\cs_new_nopar:Npn \poly_print:N #1 {
  \int_gset:Nn \l__poly_deg_int {\seq_count:N #1}
  \int_gdecr:N \l__poly_deg_int
  \int_gset:Nn \l__poly_tmpa_int {\l__poly_deg_int -
    \l__poly_trailing_int+1}
  \bool_gset_eq:NN \l__poly_ptrailing_bool \l__poly_trailing_bool
  \bool_gset_true:N \l__poly_first_bool
  \int_compare:nT {\l__poly_deg_int < \l__poly_pad_int} {
    \prg_replicate:nn {2*(\l__poly_pad_int -
      \l__poly_deg_int)}{\tl_use:N \l__poly_sep_tl}
  }
  \seq_map_inline:Nn #1 {
    \bool_if:nTF {\int_compare_p:n {##1 == 0} && \l__poly_first_bool}
    {
      \tl_use:N \l__poly_sep_tl
      \tl_use:N \l__poly_sep_tl
    }
    {
      \bool_if:nTF {\int_compare_p:n {##1 != 0} || \l__poly_zeros_bool}
      {
    \int_compare:nTF {##1 < 0} 
    {
          \bool_if:NF \l__poly_first_bool {
        \tl_use:N \l__poly_sep_tl
          }
          - \tl_use:N \l__poly_sep_tl
      \bool_if:nF {\int_compare_p:n {##1 == -1} && \int_compare_p:n {\l__poly_deg_int > 0}}
      {
        \int_eval:n {-##1}
      }
    }
    {
      \bool_if:NF \l__poly_first_bool {\tl_use:N \l__poly_sep_tl+} \tl_use:N \l__poly_sep_tl
      \bool_if:nF {\int_compare_p:n {##1 == 1} && \int_compare_p:n {\l__poly_deg_int > 0}}
      {
        ##1
      }
    }
    \int_compare:nT {\l__poly_deg_int > 0}
    {
      \tl_use:N \l__poly_var_tl
      \int_compare:nT {\l__poly_deg_int > 1} {^{\int_use:N \l__poly_deg_int}}
    }
      }
      {
    \tl_use:N \l__poly_sep_tl
    \tl_use:N \l__poly_sep_tl
      }
      \bool_gset_false:N \l__poly_first_bool
    }
    \int_gdecr:N \l__poly_deg_int
    \bool_if:nT {\l__poly_ptrailing_bool && \int_compare_p:n {\l__poly_deg_int < \l__poly_tmpa_int}} {
      \seq_map_break:
    }
  }
}
\cs_generate_variant:Nn \poly_print:N {c}

\cs_new_nopar:Npn \poly_add:NNN #1#2#3 {
  \seq_clear_new:N #1
  \int_step_inline:nnnn {1} {1} {\int_max:nn {\seq_count:N #2} {\seq_count:N #3}} {
    \seq_put_left:Nx #1 {\int_eval:n {\seq_item:Nn #2 { - ##1} + \seq_item:Nn #3 { - ##1}+0}}
  }
}
\cs_generate_variant:Nn \poly_add:NNN {Ncc,ccc}
\cs_new_nopar:Npn \poly_sub:NNN #1#2#3 {
  \seq_clear_new:N #1
  \int_step_inline:nnnn {1} {1} {\int_max:nn {\seq_count:N #2} {\seq_count:N #3}} {
    \seq_put_left:Nx #1 {\int_eval:n {\seq_item:Nn #2 { - ##1} - \seq_item:Nn #3 { - ##1}+0}}
  }
}
\cs_generate_variant:Nn \poly_sub:NNN {Ncc,ccc}
\cs_new_nopar:Npn \poly_shift:Nn #1#2 {
  \prg_replicate:nn {#2} {
    \seq_put_right:Nn #1 {0}
  }
}
\cs_new_nopar:Npn \poly_mul:NNN #1#2#3 {
  \seq_clear_new:N #1
  \group_begin:
  \seq_clear_new:N \l__poly_tmpa_seq    
  \seq_clear_new:N \l__poly_tmpb_seq    
  \seq_clear_new:N \l__poly_tmpc_seq    
  \int_set:Nn \l__poly_tmpa_int {\seq_count:N #2 - 1}
  \seq_map_inline:Nn #2 {
    \seq_clear:N \l__poly_tmpa_seq
    \seq_map_inline:Nn #3 {
      \seq_put_right:Nx \l__poly_tmpa_seq {\int_eval:n {##1 * ####1}}
    }
    \poly_shift:Nn \l__poly_tmpa_seq {\l__poly_tmpa_int}
    \poly_add:NNN \l__poly_tmpc_seq \l__poly_tmpb_seq \l__poly_tmpa_seq
    \seq_set_eq:NN \l__poly_tmpb_seq \l__poly_tmpc_seq
    \int_decr:N \l__poly_tmpa_int
  }
  \seq_gset_eq:NN \l__poly_gtmpa_seq \l__poly_tmpb_seq
  \group_end:
  \seq_set_eq:NN #1 \l__poly_gtmpa_seq
  \seq_clear:N \l__poly_gtmpa_seq
}
\cs_generate_variant:Nn \poly_mul:NNN {Ncc, ccc}
\cs_new_nopar:Npn \poly_div:NNN #1#2#3 {
  \seq_clear_new:N #1
  \seq_put_left:Nx #1 {\int_eval:n {\seq_item:Nn #2 {1} / \seq_item:Nn #3 {1}}}
  \poly_shift:Nn #1 {\seq_count:N #2 - \seq_count:N #3}
}
\cs_generate_variant:Nn \poly_div:NNN {Ncc, ccc}
\prg_new_conditional:Npnn \poly_is_divisible:NN #1#2 {p,T,F,TF} {
  \int_compare:nTF {\seq_count:N #1 < \seq_count:N #2}
  {
    \prg_return_false:
  }
  {
    \prg_return_true:
  }
}
\cs_new_nopar:Npn \poly_trim:N #1 {
  \bool_do_while:nn {\int_compare_p:n {\seq_item:Nn #1 {1} == 0}} {
    \seq_pop_left:NN #1 \l_tmpa_tl
  }
}
\cs_new_nopar:Npn \poly_longdiv:NN #1#2 {
  \group_begin:
  \seq_clear_new:N \l__poly_quotient_seq
  \seq_clear_new:N \l__poly_remainder_seq
  \seq_clear_new:N \l__poly_factor_seq
  \seq_set_eq:NN \l__poly_remainder_seq #1
  \seq_clear_new:N \l__poly_lines_seq
  \int_zero:N \l__poly_cstage_int
  \bool_do_while:nn {
    \poly_is_divisible_p:NN \l__poly_remainder_seq #2
    &&
    (\l__poly_stage_bool || \int_compare_p:n {\l__poly_stage_int > \l__poly_cstage_int})
  }
  {
    \poly_div:NNN \l__poly_factor_seq \l__poly_remainder_seq #2
    \poly_add:NNN \l__poly_tmpa_seq \l__poly_factor_seq \l__poly_quotient_seq
    \seq_set_eq:NN \l__poly_quotient_seq \l__poly_tmpa_seq
    \poly_mul:NNN \l__poly_tmpa_seq \l__poly_factor_seq #2
    \seq_put_right:NV \l__poly_lines_seq \l__poly_tmpa_seq
    \int_incr:N \l__poly_cstage_int

    \bool_if:nT {\l__poly_stage_bool || \int_compare_p:n
      {\l__poly_stage_int > \l__poly_cstage_int}}
    {
      \poly_sub:NNN \l__poly_tmpb_seq \l__poly_remainder_seq \l__poly_tmpa_seq
      \seq_set_eq:NN \l__poly_remainder_seq \l__poly_tmpb_seq
      \poly_trim:N \l__poly_remainder_seq
      \seq_put_right:NV \l__poly_lines_seq \l__poly_remainder_seq
      \int_incr:N \l__poly_cstage_int
    }
  }
  \int_set:Nn \l__poly_pad_int {\seq_count:N #1 + \seq_count:N
    #2-1}
  \tl_set:Nn \l__poly_sep_tl {&}
  \tl_set:Nn \l__poly_longdiv_sep_tl {\cr}
  \bool_set_true:N \l__poly_zeros_bool
  \int_set:Nn \l__poly_tmpa_int {2*\seq_count:N #1+1}
  \tl_set:Nn \l__poly_oline_tl {\multispan}
  \tl_put_right:Nx \l__poly_oline_tl {{\int_use:N \l__poly_tmpa_int}}
  \tl_put_right:Nn \l__poly_oline_tl {\hrulefill\cr}
  \tl_set:Nn \l__poly_uline_tl {\multispan}
  \tl_put_right:Nx \l__poly_uline_tl {{\int_eval:n {2*\seq_count:N #2 - 1}}}
  \tl_put_right:Nn \l__poly_uline_tl {\hrulefill\cr}
  \int_set:Nn \l__poly_trailing_int {\seq_count:N #2}
  \leavevmode\vbox{\halign {  $##$&&$\>##$ \crcr
    &
    \bool_if:NTF \l__poly_stage_bool
    {
      \bool_set_false:N \l__poly_trailing_bool
    }
    {
      \bool_set_true:N \l__poly_trailing_bool
      \int_set:Nn \l__poly_trailing_int {\l__poly_stage_int/2}
    }
    \poly_print:N \l__poly_quotient_seq
    \tl_use:N \l__poly_longdiv_sep_tl
    \noalign{\vskip-\normalbaselineskip\vskip\jot}
    \prg_replicate:nn {2*\seq_count:N #2} {&}
    \tl_use:N \l__poly_oline_tl
    \int_set:Nn \l__poly_pad_int {0}
    \bool_set_true:N \l__poly_trailing_bool
    \poly_print:N #2
    &
    \smash{\Bigr)}
    &
    \int_set:Nn \l__poly_pad_int {0}
    \bool_set_false:N \l__poly_trailing_bool
    \poly_print:N #1
    \tl_use:N \l__poly_longdiv_sep_tl
    \int_gzero:N \l__poly_tmpb_int
    \seq_map_inline:Nn \l__poly_lines_seq {
      \tl_gset:Nn \l__poly_tmpa_seq {##1}
      \int_gincr:N \l__poly_tmpb_int
      &
      \bool_set_true:N \l__poly_trailing_bool
      \poly_print:N \l__poly_tmpa_seq
      \bool_if:nT {\int_compare_p:n
        {\int_mod:nn{\l__poly_tmpb_int}{2} == 1} &&
        \int_compare_p:n {
          \l__poly_tmpb_int < 2*(\seq_count:N #1 - \seq_count:N #2)
        }
        &&
        \int_compare_p:n {
          \l__poly_tmpb_int != \seq_count:N \l__poly_lines_seq
        }
      } {
        &&\hfill\downarrow\hfill
      }
      \tl_use:N \l__poly_longdiv_sep_tl
      \int_compare:nT {\int_mod:nn{\l__poly_tmpb_int}{2} == 1} {
        \noalign{\vskip-\normalbaselineskip\vskip\jot}
        \prg_replicate:nn {2*\seq_count:N #2 + \l__poly_tmpb_int + 1} {&}
        \tl_use:N \l__poly_uline_tl
      }
    }
    \cr
  }}
  \group_end:
}
\cs_generate_variant:Nn \poly_longdiv:NN {cc}
\NewDocumentCommand \PolyPrint { O{} m } {
  \group_begin:
  \keys_set:nn { polynomial }
  {
    #1
  }
  \poly_print:c {polynomial #2}
  \group_end:
}
\NewDocumentCommand \PolySet { m m } {
  \seq_set_from_clist:cn {polynomial #1} {#2}
}
\NewDocumentCommand \PolyLongDiv {O{} m m } {
  \group_begin:
  \keys_set:nn { polynomial }
  {
    #1
  }
  \poly_longdiv:cc {polynomial #2} {polynomial #3}
  \group_end:
}
\ExplSyntaxOff

Then the following:

\documentclass{article}
%\url{http://tex.stackexchange.com/q/79411/86}
\usepackage{polydiv}

\begin{document}
\PolySet{a}{1,-12,0,-42}
\PolySet{b}{1,-3}
\(\PolyLongDiv{a}{b}\)
\end{document}

produces:

Any chance of implementing stage from polynom? Your version uses a better display style. — rhody, Jul 24 '18 at 00:48
Any chance you can use the package intcalc to calculate coefficients modulo a prime (for examples over the integers mod p) at each step? Signed, a pressed-for-time professor who took a look at the guts of polynom and gave up! :) — Courtney Gibbons, Mar 04 '19 at 21:35
@CourtneyGibbons Have you had a look at https://tex.stackexchange.com/a/444979/86 ? With one caveat regarding division, it might do what you want. — Andrew Stacey, Mar 04 '19 at 22:05
@LoopSpace - Awesome! As long as the divisor is monic, life is good! Thanks for the link. — Courtney Gibbons, Mar 04 '19 at 23:32
@CourtneyGibbons Non-monic is on my list of "things to do". Usually stuff on that list gets implemented if there's a specific need/request. Without guaranteeing any time-frame, would that be useful? — Andrew Stacey, Mar 04 '19 at 23:34
It would be useful -- but don't put it as a high-urgency item unless you need a reason to pick something to do and nothing else is appealing! — Courtney Gibbons, Mar 04 '19 at 23:46
This is a very good solution, however the script went error when PolySet{a} is divisible by PolySet{b}. e.g. ( \PolySet{a}{2,2,-40,-100}, \PolySet{b}{1,6,10} ). Is it possible to fix this error? Thanks in advance! — Aliceyuenyuen, Sep 22 '22 at 00:28
@Aliceyuenyuen Over the years I've tweaked the code to fix stuff and the latest version is at: https://github.com/loopspace/Polynomial-division If that hasn't already fixed the issue, let me know — Andrew Stacey, Sep 22 '22 at 05:54

How can I get the ordinary polynomial long division by using polydiv like this?

1 Answers1

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