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I wanted to apply Ruffini's rule but since I'm using fractions it's all crowded and messy. Is there a way to leave more space between the lines?

\begin{equation*}
    \begin{array}{c|c c c|c}
        &1 &-2  &1 &-\frac{4}{27}\\
        \frac{1}{3} & &\frac{1}{3} &-\frac{5}{9} &\frac{4}{27}\\
        \hline
        & 1 &-\frac{5}{3} &\frac{4}{9} &0\\
    \end{array}
\end{equation*}
  • This seems a duplicate of tex.stackexchange.com/q/122699/4427 I'll add a variation of my solution to be more “Italian style”. – egreg Apr 20 at 13:36
  • @egreg; I don't think the link tackles the specific problem of fractions in tables: the O.P. said it's all crowded, which seems to mean fractions touch the horizontal lines. – Bernard Apr 20 at 13:49
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A solution with the cellspace package, to add a minimal padding at the top and bottom of cells in columns with specifier prefixed with the letter S (or C if you load siunitx). I also loaded nccmath to have medium-sized fractions, which look nicer in this context, from my point of view:

\documentclass[a4paper,12pt]{article}
\usepackage{cellspace}
\setlength{\cellspacetoplimit}{4pt}
\setlength{\cellspacebottomlimit}{4pt}

\usepackage{amsmath, nccmath}

\begin{document}

\[ \begin{array}{>{$}Sr<{$}|*{3}{>{$}Sr<{$}}|>{$}Sr<{$}}
    &1 &-2 &1 &-\mfrac{4}{27} \\
    \mfrac{1}{3} & &\mfrac{1}{3} &-\mfrac{5}{9} &\mfrac{4}{27} \\
    \hline
    & 1 &-\mfrac{5}{3} &\mfrac{4}{9} & 0 
\end{array} \]

\end{document} 

enter image description here

1

A variation on my solution to Command for the table Ruffini-Horner algorithm

The optional argument to \ruffini sets gapes for the cells when you have to accommodate large objects

\documentclass{article}
\usepackage{xparse,array,makecell}

\ExplSyntaxOn

\NewDocumentCommand{\ruffini}{O{0pt}mmmm}
 {% #1 = gape, #2 = polynomial, #3 = divisor, #4 = middle row, #5 = result
  \franklin_ruffini:nnnnn { #2 } { #3 } { #4 } { #5 } { #1 }
 }

\seq_new:N \l_franklin_temp_seq
\tl_new:N \l_franklin_scheme_tl
\int_new:N \l_franklin_degree_int

\cs_new_protected:Npn \franklin_ruffini:nnnnn #1 #2 #3 #4 #5
 {
  \group_begin:
  \makegapedcells
  \setcellgapes{#5}
  % Start the first row
  \tl_set:Nn \l_franklin_scheme_tl { & }
  % Split the list of coefficients
  \seq_set_split:Nnn \l_franklin_temp_seq { , } { #1 }
  % Remember the number of columns
  \int_set:Nn \l_franklin_degree_int { \seq_count:N \l_franklin_temp_seq }
  % Fill the first row
  \tl_put_right:Nx \l_franklin_scheme_tl
   { \seq_use:Nn \l_franklin_temp_seq { & } }
  % End the first row and leave two empty places in the next
  \tl_put_right:Nn \l_franklin_scheme_tl { \\ #2 & & }
  % Split the list of coefficients and fill the second row
  \seq_set_split:Nnn \l_franklin_temp_seq { , } { #3 }
  \tl_put_right:Nx \l_franklin_scheme_tl
   { \seq_use:Nn \l_franklin_temp_seq { & } }
  % End the second row
  \tl_put_right:Nn \l_franklin_scheme_tl { \\ \hline }
  % Split and fill the third row
  \seq_set_split:Nnn \l_franklin_temp_seq { , } { #4 }
  \tl_put_right:Nx \l_franklin_scheme_tl
   { & \seq_use:Nn \l_franklin_temp_seq { & } }
  % Start the array (with \use:x because the array package
  % doesn't expand the argument)
  \use:x
   {
    \exp_not:n { \begin{array} } { r | *{\int_eval:n { \l_franklin_degree_int - 1 }} { r } | r }
   }
  % Body of the array and finish
  \tl_use:N \l_franklin_scheme_tl
  \end{array}
  \group_end:
 }
\ExplSyntaxOff

\begin{document}

\[
\ruffini[2pt]{1,-2,1,-\frac{4}{27}}
  {\frac{1}{3}}
  {\frac{1}{3},-\frac{5}{9},\frac{4}{27}}
  {1,-\frac{5}{3},\frac{4}{9},0}
\qquad
\ruffini{1,-6,11,-6}{2}{2,-8,6}{1,-4,3,0}
\]

\end{document}

enter image description here

1

with use of makecell:

\documentclass[a4paper,12pt]{article}
\usepackage{makecell}
\usepackage{nccmath}

\begin{document}
\[\setcellgapes{3pt}
  \makegapedcells
\begin{array}{r|*{3}{r}|r}
                &   1   &   -2              &   1               &   -\mfrac{4}{27}  \\
\mfrac{1}{3}    &       &    \mfrac{1}{3}   &   -\mfrac{5}{9}   &   \mfrac{4}{27}   \\
    \hline
                & 1     &   -\mfrac{5}{3}   &   \mfrac{4}{9}    &   0
\end{array}
\]
\end{document}

enter image description here

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