%This command provides the math for the second horizontal area
%of the first column of the page 6
%
%The command has one parameter
%       1) The width of the math text
\newcommand\TSixPartialFrac[1]{%
   \parbox[t]{#1}{%
      \TSixPartialFontSize
      \DisplaySpace{\TSixDisplaySpace}{\TSixDisplayShortSpace}
      %Since the columns is narrow, ragged right looks better
      \raggedright

      Let $N(x)$ and $D(x)$ be polynomial functions of $x$.

      We can break down $N(x)/D(x)$ using partial fraction expansion.

      First,
      if the degree of $N$ is greater than or equal to the degree of $D$,
      divide $N$ by $D$,
      obtaining
      \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
      \Fm{\frac{N(x)}{D(x)} = Q(x) + \frac{N'(x)}{D(x)}}
      \end{DisplayFormulae}
      where the degree of $N'$ is less than that of $D$.

      \TSixTitle{Second, factor $D(x)$}
      Use the following rules:

      \mbox{For a non-repeated factor:}
           \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
           \Fm{\frac{N(x)}{(x-a) D(x)} = \frac{A}{x-a} + \frac{N'(x)}{D(x)}}
           where
           \Fm{A = \left[\frac{N(x)}{D(x)}\right]_{x=a}}
           \end{DisplayFormulae}

      \mbox{For a repeated factor:}
          \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
          \Fm{\frac{N(x)}{(x-a)^m D(x)} =\sum_{k=0}^{m-1}\frac{A_k}{(x-a)^{m-k}} + \frac{N'(x)}{D(x)}}
          where
          \Fm{A_k = \frac{1}{k!}\left[\frac{d^k}{dx^k} 
                    \left(\frac{N(x)}{D(x)}\right)\right]_{x=a}}
          \end{DisplayFormulae}
   }
}
