7

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 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 1 '12 at 6:21
  • If you're looking for the 0x terms, then see this answer – Scott H. Nov 1 '12 at 6:25
  • 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 1 '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
6

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:

polynomial long division

  • Any chance of implementing stage from polynom? Your version uses a better display style. – rhody Jul 24 '18 at 0:48
  • 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 4 at 21:35
  • @CourtneyGibbons Have you had a look at tex.stackexchange.com/a/444979/86 ? With one caveat regarding division, it might do what you want. – Loop Space Mar 4 at 22:05
  • @LoopSpace - Awesome! As long as the divisor is monic, life is good! Thanks for the link. – Courtney Gibbons Mar 4 at 23:32
  • 1
    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 4 at 23:46

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