\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  - 2x  - 2x  + 4x  + 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  - 2x  - 2x  + 4x  + 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            7
       - 544320x   + 4354560x  - 14696640x  + 28615680x
     + 
                  6            5            4            3
       - 40085280x  + 46656000x  - 39411360x  + 18247680x
     + 
                 2
       - 5870880x  + 3317760x + 246240
  /
        20      19      18      17       16       15
       x   - 12x   + 53x   - 76x   - 159x   + 676x
     + 
             14        13        12        11        10
       - 391x   - 1596x   + 2527x   + 1148x   - 4977x
     + 
            9        8        7        6        5       4
       1372x  + 4907x  - 3444x  - 2381x  + 2924x  + 276x
     + 
              3       2
       - 1184x  + 208x  + 192x - 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  - 4x  + 3x  + 9x  - 6x  - 4x - 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
           2x  - 7x  + 26x  + 8x
   (11)  ------------------------
          8     6     4     2
         x  - 5x  + 6x  + 4x  - 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      15
       x   + 2x   + 4x   + 7x   + 10x   + 17x   + 22x
     + 
          14      13      12      11      10      9      8
       30x   + 36x   + 40x   + 47x   + 46x   + 49x  + 43x
     + 
          7      6      5      4      3     2
       38x  + 32x  + 23x  + 19x  + 10x  + 7x  + 2x + 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}

