\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: Domain \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) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴 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 컴� - 3%A - 6 (3) 컴컴� - 컴컴� + > 컴컴컴컴� x - 2 x + 1 컴� 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) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴 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 컴� 2160%A + 4320 - 컴컴컴컴 + 컴컴컴컴 + > 컴컴컴컴컴컴� 6 6 컴� 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) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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 컴컴 컴� 컴 2401 343 49 컴컴컴 + 컴컴컴컴 + 컴컴컴컴 x - 2 2 3 (x - 2) (x - 2) + 179 135 - 컴컴 %A + 컴컴 컴� 2401 2401 > 컴컴컴컴컴컴컴컴 컴� x - %A 2 %A + %A + 1= 0 + 37 20 컴컴 %A + 컴컴 컴� 1029 1029 > 컴컴컴컴컴컴컴 컴� 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) 컴컴컴컴컴컴컴컴컴컴컴컴 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 � 컴� 2 컴� 1 > 컴컴컴 + > 컴컴컴컴� 컴� x - %A 컴� 3 2 2 (x - %A) %A - 2= 0 %A - 2= 0 + 1 � 컴� 2 > 컴컴컴 컴� 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 - 컴 컴� 2 컴� 9 27 > 컴컴컴 + > 컴컴컴컴� 컴� x - %A 컴� x - %A 2 2 %A + 1= 0 %A + %A + 1= 0 + 1 1 컴 %A - 컴 컴� 27 27 > 컴컴컴컴컴 컴� 2 2 (x - %A) %A + %A + 1= 0 + SIGMA 5 2 %A + %A + 1= 0 , 96556567040 4 420961732891 3 - 컴컴컴컴컴컴 %A + 컴컴컴컴컴컴 %A 912390759099 912390759099 + 59101056149 2 373545875923 - 컴컴컴컴컴컴 %A - 컴컴컴컴컴컴 %A 912390759099 912390759099 + 529673492498 컴컴컴컴컴컴 912390759099 / x - %A + SIGMA 5 2 %A + %A + 1= 0 , 5580868 4 2024443 3 4321919 2 - 컴컴컴컴 %A - 컴컴컴컴 %A + 컴컴컴컴 %A 94070601 94070601 94070601 + 84614 5070620 - 컴컴컴� %A - 컴컴컴컴 1542141 94070601 / 2 (x - %A) + SIGMA 5 2 %A + %A + 1= 0 , 1610957 4 2763014 3 2016775 2 컴컴컴컴 %A + 컴컴컴컴 %A - 컴컴컴컴 %A 94070601 94070601 94070601 + 266953 4529359 컴컴컴컴 %A + 컴컴컴컴 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}