\begin{patch}{FullPartialFractionExpansionXmpPagePatch1}
\begin{paste}{FullPartialFractionExpansionXmpPageFull1}{FullPartialFractionExpansionXmpPageEmpty1}
\pastebutton{FullPartialFractionExpansionXmpPageFull1}{\hidepaste}
\tab{5}\spadcommand{Fx := FRAC UP(x, FRAC INT)\bound{Fx }}
\indentrel{3}\begin{verbatim}
   (1)
   Fraction(UnivariatePolynomial(x,Fraction(Integer)))
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty1}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty1}{FullPartialFractionExpansionXmpPagePatch1}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{Fx := FRAC UP(x, FRAC INT)\bound{Fx }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch2}
\begin{paste}{FullPartialFractionExpansionXmpPageFull2}{FullPartialFractionExpansionXmpPageEmpty2}
\pastebutton{FullPartialFractionExpansionXmpPageFull2}{\hidepaste}
\tab{5}\spadcommand{f : Fx := 36 / (x^5-2*x^4-2*x^3+4*x^2+x-2)\bound{f }\free{Fx }}
\indentrel{3}\begin{verbatim}
                       36
   (2)  -------------------------------
         5      4      3      2
        x  - 2 x  - 2 x  + 4 x  + x - 2
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty2}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty2}{FullPartialFractionExpansionXmpPagePatch2}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{f : Fx := 36 / (x^5-2*x^4-2*x^3+4*x^2+x-2)\bound{f }\free{Fx }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch3}
\begin{paste}{FullPartialFractionExpansionXmpPageFull3}{FullPartialFractionExpansionXmpPageEmpty3}
\pastebutton{FullPartialFractionExpansionXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{g := fullPartialFraction f\bound{g }\free{f }}
\indentrel{3}\begin{verbatim}
          4       4         --+      - 3 %A - 6
   (3)  ----- - ----- +     >        ----------
        x - 2   x + 1       --+               2
                          2           (x - %A)
                        %A  - 1 = 0
Type: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty3}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty3}{FullPartialFractionExpansionXmpPagePatch3}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{g := fullPartialFraction f\bound{g }\free{f }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch4}
\begin{paste}{FullPartialFractionExpansionXmpPageFull4}{FullPartialFractionExpansionXmpPageEmpty4}
\pastebutton{FullPartialFractionExpansionXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{g :: Fx\free{g }}
\indentrel{3}\begin{verbatim}
                       36
   (4)  -------------------------------
         5      4      3      2
        x  - 2 x  - 2 x  + 4 x  + x - 2
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty4}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty4}{FullPartialFractionExpansionXmpPagePatch4}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{g :: Fx\free{g }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch5}
\begin{paste}{FullPartialFractionExpansionXmpPageFull5}{FullPartialFractionExpansionXmpPageEmpty5}
\pastebutton{FullPartialFractionExpansionXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{g5 := D(g, 5)\free{g }\bound{g5 }}
\indentrel{3}\begin{verbatim}
   (5)
        480        480         --+      2160 %A + 4320
   - -------- + -------- +     >        --------------
            6          6       --+                 7
     (x - 2)    (x + 1)      2             (x - %A)
                           %A  - 1 = 0
Type: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty5}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty5}{FullPartialFractionExpansionXmpPagePatch5}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{g5 := D(g, 5)\free{g }\bound{g5 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch6}
\begin{paste}{FullPartialFractionExpansionXmpPageFull6}{FullPartialFractionExpansionXmpPageEmpty6}
\pastebutton{FullPartialFractionExpansionXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{f5 := D(f, 5)\free{f }\bound{f5 }}
\indentrel{3}\begin{verbatim}
   (6)
                 10            9             8
       - 544320 x   + 4354560 x  - 14696640 x
     + 
                 7             6             5
       28615680 x  - 40085280 x  + 46656000 x
     + 
                   4             3            2
       - 39411360 x  + 18247680 x  - 5870880 x
     + 
       3317760 x + 246240
  /
        20       19       18       17        16        15
       x   - 12 x   + 53 x   - 76 x   - 159 x   + 676 x
     + 
              14         13         12         11
       - 391 x   - 1596 x   + 2527 x   + 1148 x
     + 
               10         9         8         7         6
       - 4977 x   + 1372 x  + 4907 x  - 3444 x  - 2381 x
     + 
             5        4         3        2
       2924 x  + 276 x  - 1184 x  + 208 x  + 192 x - 64
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty6}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty6}{FullPartialFractionExpansionXmpPagePatch6}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{f5 := D(f, 5)\free{f }\bound{f5 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch7}
\begin{paste}{FullPartialFractionExpansionXmpPageFull7}{FullPartialFractionExpansionXmpPageEmpty7}
\pastebutton{FullPartialFractionExpansionXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{g5::Fx - f5\free{Fx g5 f5 }}
\indentrel{3}\begin{verbatim}
   (7)  0
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty7}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty7}{FullPartialFractionExpansionXmpPagePatch7}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{g5::Fx - f5\free{Fx g5 f5 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch8}
\begin{paste}{FullPartialFractionExpansionXmpPageFull8}{FullPartialFractionExpansionXmpPageEmpty8}
\pastebutton{FullPartialFractionExpansionXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{f : Fx := (x^5 * (x-1)) / ((x^2 + x + 1)^2 * (x-2)^3)\free{Fx }\bound{f2 }}
\indentrel{3}\begin{verbatim}
                          6    5
                         x  - x
   (8)  ----------------------------------------
         7      6      5      3      2
        x  - 4 x  + 3 x  + 9 x  - 6 x  - 4 x - 8
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty8}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty8}{FullPartialFractionExpansionXmpPagePatch8}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{f : Fx := (x^5 * (x-1)) / ((x^2 + x + 1)^2 * (x-2)^3)\free{Fx }\bound{f2 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch9}
\begin{paste}{FullPartialFractionExpansionXmpPageFull9}{FullPartialFractionExpansionXmpPageEmpty9}
\pastebutton{FullPartialFractionExpansionXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{g := fullPartialFraction f\free{f2 }\bound{g2 }}
\indentrel{3}\begin{verbatim}
   (9)
      1952       464        32
      ----       ---        --
      2401       343        49
     ------ + -------- + --------
      x - 2          2          3
              (x - 2)    (x - 2)
   + 
                          179       135
                       - ---- %A + ----
           --+           2401      2401
           >           ----------------
           --+              x - %A
       2
     %A  + %A + 1 = 0
   + 
                        37        20
                       ---- %A + ----
           --+         1029      1029
           >           --------------
           --+                    2
       2                  (x - %A)
     %A  + %A + 1 = 0
Type: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty9}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty9}{FullPartialFractionExpansionXmpPagePatch9}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{g := fullPartialFraction f\free{f2 }\bound{g2 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch10}
\begin{paste}{FullPartialFractionExpansionXmpPageFull10}{FullPartialFractionExpansionXmpPageEmpty10}
\pastebutton{FullPartialFractionExpansionXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{g :: Fx - f\free{f2 g2 Fx }}
\indentrel{3}\begin{verbatim}
   (10)  0
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty10}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty10}{FullPartialFractionExpansionXmpPagePatch10}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{g :: Fx - f\free{f2 g2 Fx }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch11}
\begin{paste}{FullPartialFractionExpansionXmpPageFull11}{FullPartialFractionExpansionXmpPageEmpty11}
\pastebutton{FullPartialFractionExpansionXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{f : Fx := (2*x^7-7*x^5+26*x^3+8*x) / (x^8-5*x^6+6*x^4+4*x^2-8)\free{Fx }\bound{f3 }}
\indentrel{3}\begin{verbatim}
             7      5       3
          2 x  - 7 x  + 26 x  + 8 x
   (11)  ---------------------------
          8      6      4      2
         x  - 5 x  + 6 x  + 4 x  - 8
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty11}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty11}{FullPartialFractionExpansionXmpPagePatch11}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{f : Fx := (2*x^7-7*x^5+26*x^3+8*x) / (x^8-5*x^6+6*x^4+4*x^2-8)\free{Fx }\bound{f3 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch12}
\begin{paste}{FullPartialFractionExpansionXmpPageFull12}{FullPartialFractionExpansionXmpPageEmpty12}
\pastebutton{FullPartialFractionExpansionXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{g := fullPartialFraction f\free{f3 }\bound{g3 }}
\indentrel{3}\begin{verbatim}
   (12)
                     1
                     -
         --+         2         --+          1
         >        ------ +     >        ---------
         --+      x - %A       --+              3
       2                     2          (x - %A)
     %A  - 2 = 0           %A  - 2 = 0
   + 
                     1
                     -
         --+         2
         >        ------
         --+      x - %A
       2
     %A  + 1 = 0
Type: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty12}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty12}{FullPartialFractionExpansionXmpPagePatch12}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{g := fullPartialFraction f\free{f3 }\bound{g3 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch13}
\begin{paste}{FullPartialFractionExpansionXmpPageFull13}{FullPartialFractionExpansionXmpPageEmpty13}
\pastebutton{FullPartialFractionExpansionXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{g :: Fx - f\free{f3 g3 Fx }}
\indentrel{3}\begin{verbatim}
   (13)  0
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty13}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty13}{FullPartialFractionExpansionXmpPagePatch13}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{g :: Fx - f\free{f3 g3 Fx }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch14}
\begin{paste}{FullPartialFractionExpansionXmpPageFull14}{FullPartialFractionExpansionXmpPageEmpty14}
\pastebutton{FullPartialFractionExpansionXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{f:Fx := x^3 / (x^21 + 2*x^20 + 4*x^19 + 7*x^18 + 10*x^17 + 17*x^16 + 22*x^15 + 30*x^14 + 36*x^13 + 40*x^12 + 47*x^11 + 46*x^10 + 49*x^9 + 43*x^8 + 38*x^7 + 32*x^6 + 23*x^5 + 19*x^4 + 10*x^3 + 7*x^2 + 2*x + 1)\free{Fx }\bound{f4 }}
\indentrel{3}\begin{verbatim}
   (14)
      3
     x
  /
        21      20      19      18       17       16
       x   + 2 x   + 4 x   + 7 x   + 10 x   + 17 x
     + 
           15       14       13       12       11       10
       22 x   + 30 x   + 36 x   + 40 x   + 47 x   + 46 x
     + 
           9       8       7       6       5       4
       49 x  + 43 x  + 38 x  + 32 x  + 23 x  + 19 x
     + 
           3      2
       10 x  + 7 x  + 2 x + 1
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty14}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty14}{FullPartialFractionExpansionXmpPagePatch14}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{f:Fx := x^3 / (x^21 + 2*x^20 + 4*x^19 + 7*x^18 + 10*x^17 + 17*x^16 + 22*x^15 + 30*x^14 + 36*x^13 + 40*x^12 + 47*x^11 + 46*x^10 + 49*x^9 + 43*x^8 + 38*x^7 + 32*x^6 + 23*x^5 + 19*x^4 + 10*x^3 + 7*x^2 + 2*x + 1)\free{Fx }\bound{f4 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch15}
\begin{paste}{FullPartialFractionExpansionXmpPageFull15}{FullPartialFractionExpansionXmpPageEmpty15}
\pastebutton{FullPartialFractionExpansionXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{g := fullPartialFraction f\free{f4 }\bound{g4 }}
\indentrel{3}\begin{verbatim}
   (15)
                   1                         1      19
                   - %A                      - %A - --
         --+       2             --+         9      27
         >        ------ +       >           ---------
         --+      x - %A         --+           x - %A
       2                     2
     %A  + 1 = 0           %A  + %A + 1 = 0
   + 
                        1       1
                       -- %A - --
           --+         27      27
           >           ----------
           --+                  2
       2                (x - %A)
     %A  + %A + 1 = 0
   + 
     SIGMA
          5     2
        %A  + %A  + 1 = 0
    ,
               96556567040   4   420961732891   3
            - ------------ %A  + ------------ %A
              912390759099       912390759099
          + 
               59101056149   2   373545875923
            - ------------ %A  - ------------ %A
              912390759099       912390759099
          + 
            529673492498
            ------------
            912390759099
       /
          x - %A
   + 
     SIGMA
          5     2
        %A  + %A  + 1 = 0
    ,
               5580868   4    2024443   3    4321919   2
            - -------- %A  - -------- %A  + -------- %A
              94070601       94070601       94070601
          + 
               84614        5070620
            - ------- %A - --------
              1542141      94070601
       /
                  2
          (x - %A)
   + 
     SIGMA
          5     2
        %A  + %A  + 1 = 0
    ,
             1610957   4    2763014   3    2016775   2
            -------- %A  + -------- %A  - -------- %A
            94070601       94070601       94070601
          + 
             266953        4529359
            -------- %A + --------
            94070601      94070601
       /
                  3
          (x - %A)
Type: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty15}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty15}{FullPartialFractionExpansionXmpPagePatch15}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{g := fullPartialFraction f\free{f4 }\bound{g4 }}
\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPagePatch16}
\begin{paste}{FullPartialFractionExpansionXmpPageFull16}{FullPartialFractionExpansionXmpPageEmpty16}
\pastebutton{FullPartialFractionExpansionXmpPageFull16}{\hidepaste}
\tab{5}\spadcommand{g :: Fx - f\free{f4 g4 Fx }}
\indentrel{3}\begin{verbatim}
   (16)  0
Type: Fraction(UnivariatePolynomial(x,Fraction(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{FullPartialFractionExpansionXmpPageEmpty16}
\begin{paste}{FullPartialFractionExpansionXmpPageEmpty16}{FullPartialFractionExpansionXmpPagePatch16}
\pastebutton{FullPartialFractionExpansionXmpPageEmpty16}{\showpaste}
\tab{5}\spadcommand{g :: Fx - f\free{f4 g4 Fx }}
\end{paste}\end{patch}

