{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart: with(algcur ves):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "The puiseux program can give unnecessarily large answers. Reading the following code into Map le before using the puiseux command will help to prevent one (but not \+ all) of the causes of these large answers, namely it will help to prev ent some unnecessary expanding." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2225 "`algcurves/lift_exp` := proc(v, f, x, y)\nlocal i, ii, r, re s, v7, vv7, v3, ext, a, j, n, np, ram, j3;\n if v[5] = 1 then retur n \{v\} end if;\n v3 := degree(v[3], x);\n res := \{\};\n r : = v[1] + y*x^v[2];\n vv7 := v[7]*v3 + v[2] - 1;\n vv7 := vv7 + v [5];\n ii := `algcurves/truncate_subs`(subs(x = v[3], f), x, y, r, \+ vv7 + 1,\n v[4]);\n if ii = 0 then error \"degree estimate w as wrong\" end if;\n v7 := (ldegree(ii, x) - v[2])/v3;\n r := `a lgcurves/v_ext_m`(\n `algcurves/g_factors`(tcoeff(ii, x), y, v[ 4]), y);\n for i in r do res := res union `algcurves/lift_exp`([\n \+ v[1] + x^v[2]*i[1], v[2] + 1, v[3], [op(i[3]), op(v[4])], i[2], \n v[6]*i[4], v7, [op(v[8]), [op(1 .. 4, v)]]], f, x, y)\n e nd do;\n if add(i[5]*i[6]*degree(i[3], x)/(v[6]*v3), i = res) <>\n \+ degree(tcoeff(ii, x), y) then error \"found wrong number of expansi ons\"\n end if;\n if v[5] = degree(tcoeff(ii, x), y) then\n \+ if ldegree(ii, x) <> vv7 then error \"degree estimate was wrong\"\n end if;\n return res\n end if;\n ii := collect(ii , y);\n ii := add(`algcurves/normal_tcoeff`(coeff(ii, y, i), x)*y^i ,\n i = 0 .. degree(ii, y));\n np := `algcurves/Newtonpolygo n`(ii, x, y);\n if nops(np) = 2 and np[1][3] = 0 then\n erro r \"found wrong number of expansions\"\n end if;\n for j in np d o\n if 2 < nops(j) and 0 < j[3] and j[3] < 1 then\n \+ r := `algcurves/g_factors`(j[4], x, v[4]);\n r := `algcurve s/v_ext_m`(r, x);\n for i in r do\n j3 := j[ 3] - v[2];\n ext := [op(i[3]), op(v[4])];\n \+ n := mods(1/numer(j3), denom(j3));\n ram := i[1]^n* x^denom(j3);\n a := v[2]*denom(j3) - numer(j[3]);\n \+ res := res union `algcurves/lift_exp`([collect(\n \+ subs(x = ram, v[1])\n + x^a*i[1]^((1 - \+ n*numer(j3))/denom(j3)), x, normal),\n a + 1, norma l(subs(x = ram, v[3])), ext, i[2],\n v[6]*i[4],\n \+ (j[2] - j[1]*j[3] - a/degree(ram, x))/degree(v[3], x ),\n [op(v[8]), [op(1 .. 4, v)]]], f, x, y)\n \+ end do\n end if\n end do;\n res\nend proc:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "A:=(x^2+y)^2*(x^2-y)+a*x*y^ 3+b*y^4+c*x^3*y^2+d*x^2*y^3+e*x*y^4+f*y^5+g*x^5*y+h*x^4*y^2+j*x^3*y^3+ k*x^2*y^4+l*x*y^5+m*y^6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,<*& ),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-F,F-,&F)F-F.!\"\"F-F-*(%\"aGF-F+F-)F.\" \"$F-F-*&%\"bGF-)F.\"\"%F-F-*(%\"cGF-)F+F4F-)F.F,F-F-*(%\"dGF-F*F-F3F- F-*(%\"eGF-F+F-F7F-F-*&%\"fGF-)F.\"\"&F-F-*(%\"gGF-)F+FDF-F.F-F-*(%\"h GF-)F+F8F-F " 0 "" {MPLTEXT 1 0 19 "puiseux(A,x=0,y,0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%-%'RootOfG6#,**&%\"mG\"\"\")%# _ZG\"\"$F*F**&%\"fGF*)F,\"\"#F*F*F*!\"\"*&%\"bGF*F,F*F**$)%\"xGF1F*,&* (\"\")F2,(%\"gGF*%\"aGF*%\"cGF2F-*&F7F*,(*&F1F2FF*F2F2#\"\"&F1F***\"\"%F2F;F1F7F1F@!\"#F2" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 15 "numerator=0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A1:=eval(A,c=a+g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G,< *&),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-F,F-,&F)F-F.!\"\"F-F-*(%\"aGF-F+F-)F. \"\"$F-F-*&%\"bGF-)F.\"\"%F-F-*(,&F2F-%\"gGF-F-)F+F4F-)F.F,F-F-*(%\"dG F-F*F-F3F-F-*(%\"eGF-F+F-F7F-F-*&%\"fGF-)F.\"\"&F-F-*(F;F-)F+FEF-F.F-F -*(%\"hGF-)F+F8F-F=F-F-*(%\"jGF-F " 0 "" {MPLTEXT 1 0 20 "puiseux(A1,x=0,y,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%-%'RootOfG6#,**&%\"mG\"\"\" )%#_ZG\"\"$F*F**&%\"fGF*)F,\"\"#F*F*F*!\"\"*&%\"bGF*F,F*F**$)%\"xGF1F* ,&F5F2*&)F7F-F*-F%6#,,*&F1F*F0F*F**&,&%\"aGF*%\"gGF2F*F,F*F*%\"hGF*F4F *%\"dGF2F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "discrim=0 [4]" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "discrim(2*_Z^2+(a-g)*_Z+h+b -d,_Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\")\"\"\"%\"dGF&F&*&F %F&%\"hGF&!\"\"*&F%F&%\"bGF&F**(\"\"#F&%\"gGF&%\"aGF&F**$)F/F.F&F&*$)F 0F.F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,d);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\")!\"\"%\"gG\"\"#F&%\"hG\"\"\"% \"bGF**(\"\"%F&F'F*%\"aGF*F**&F%F&F.F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A2:=eval(A1,d=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%#A2G,<*&),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-F,F-,&F)F-F.!\"\"F-F-*(%\"aGF -F+F-)F.\"\"$F-F-*&%\"bGF-)F.\"\"%F-F-*(,&F2F-%\"gGF-F-)F+F4F-)F.F,F-F -*(,,*&\"\")F0F;F,F0%\"hGF-F6F-*(F8F0F;F-F2F-F-*&FAF0F2F,F0F-F*F-F3F-F -*(%\"eGF-F+F-F7F-F-*&%\"fGF-)F.\"\"&F-F-*(F;F-)F+FJF-F.F-F-*(FBF-)F+F 8F-F=F-F-*(%\"jGF-F " 0 "" {MPLTEXT 1 0 20 "puiseux(A 2,x=0,y,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,(*(\"*caVo#!\"\",6*& )%\"aG\"\"#\"\"\"%\"gGF-F'*(\"\"&F-)F.F,F-F+F-F-*$)F+\"\"$F-F'*&F4F-)F .F4F-F'*&\"#kF-%\"eGF-F-*&F8F-%\"jGF-F'*(\"#;F-%\"hGF-F+F-F'*(F=F-F>F- F.F-F-*(F=F-%\"bGF-F+F-F-*(F=F-FAF-F.F-F'\"\"%*&%\"xGF-,6*(\"$G\"F'F+F ,F.F-F-**F0F-FHF'F.F,F+F-F'*&FHF'F+F4F-*(F4F-FHF'F.F4F-*&F,F'F9F-F'*&F ,F'F;F-F-*(\"\")F'F>F-F+F-F-*(FOF'F>F-F.F-F'*(FOF'FAF-F+F-F'*(FOF'FAF- F.F-F-F'#\"\"(F,F-*,\"(3')Q)F'F(F4,&F+F-F.F'F-FEF4FF!\"$F-**\"&%Q;F'F( F,FEF,FF!\"#F'-%'RootOfG6#,**&%\"mGF-)%#_ZGF4F-F-*&%\"fGF-)F]oF,F-F-F- F'*&FAF-F]oF-F-*$)FEF,F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "numer ator=0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "solve(1/128*a ^2*g-5/128*g^2*a+1/128*a^3+3/128*g^3-1/2*e+1/2*j+1/8*h*a-1/8*h*g-1/8*b *a+1/8*b*g,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#;!\"\",2*&)% \"aG\"\"#\"\"\"%\"gGF,F&*(\"\"&F,)F-F+F,F*F,F,*$)F*\"\"$F,F&*&F3F,)F-F 3F,F&*&\"#kF,%\"eGF,F,*&F7F,%\"jGF,F&*(F%F,%\"bGF,F*F,F,*(F%F,F " 0 "" {MPLTEXT 1 0 17 "A3:=e val(A2,h=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A3G,<*&),&*$)%\"xG \"\"#\"\"\"F-%\"yGF-F,F-,&F)F-F.!\"\"F-F-*(%\"aGF-F+F-)F.\"\"$F-F-*&% \"bGF-)F.\"\"%F-F-*(,&F2F-%\"gGF-F-)F+F4F-)F.F,F-F-*(,,*&\"\")F0F;F,F0 *(\"#;F0,2*&)F2F,F-F;F-F0*(\"\"&F-)F;F,F-F2F-F-*$)F2F4F-F0*&F4F-)F;F4F -F0*&\"#kF-%\"eGF-F-*&FOF-%\"jGF-F0*(FCF-F6F-F2F-F-*(FCF-F6F-F;F-F0F-, &F2F-F;F0F0F-F6F-*(F8F0F;F-F2F-F-*&FAF0F2F,F0F-F*F-F3F-F-*(FPF-F+F-F7F -F-*&%\"fGF-)F.FHF-F-*(F;F-)F+FHF-F.F-F-*,FCF0FDF-FUF0F+F8F.F,F-*(FRF- F " 0 "" {MPLTEXT 1 0 20 "puiseux(A3,x=0,y,0);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%-%'RootOfG6#,**&%\"mG\"\"\")%#_ZG\" \"$F*F**&%\"fGF*)F,\"\"#F*F*F*!\"\"*&%\"bGF*F,F*F*,(*$)%\"xGF1F*F2*&)F 8F-F*,&*&\"\"%F2%\"aGF*F2*&F=F2%\"gGF*F*F*F**&)F8F=F*-F%6#,L*&,&*&\"$G \"F*F>F*F**&FIF*F@F*F2F*F0F*F**&,.*(\"#SF*)F@F1F*F>F*F**&\"\")F*)F@F-F *F2*&\"#CF*)F>F-F*F**&\"$c#F*%\"eGF*F**&FWF*%\"jGF*F2*(\"#cF*)F>F1F*F@ F*F2F*F,F*F**(\"#;F*FZF*FgnF*F2**\"#7F*F4F*F>F*FOF*F***\"#KF*FZF*F@F*F >F*F***F[oF*F4F*FgnF*F@F*F2*$)F>\"\"&F*F**(F=F*F4F*FUF*F**(F=F*F4F*FRF *F2*(F]oF*FXF*FgnF*F**(F]oF*FXF*FOF*F**(F=F*)F>F=F*F@F*F2*(\"\"'F*FUF* FOF*F**(F=F*FgnF*FRF*F2*&)F@F=F*F>F*F**(\"#kF*%\"kGF*F>F*F**(FinF*FZF* FOF*F2*(F^pF*F/F*F>F*F2*(F^pF*F/F*F@F*F**(F^pF*F_pF*F@F*F2**F^pF*FXF*F >F*F@F*F2F*F*F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "discrim=0 [4] " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "discrim((128*a-128*g)* _Z^2+(40*g^2*a-8*g^3+24*a^3+256*e-256*j-56*a^2*g)*_Z-16*j*a^2+12*b*a*g ^2+32*j*g*a-12*b*a^2*g+a^5+4*b*a^3-4*b*g^3+32*e*a^2+32*e*g^2-4*a^4*g+6 *a^3*g^2-4*a^2*g^3+g^4*a+64*k*a-16*j*g^2-64*f*a+64*f*g-64*k*g-64*e*a*g ,_Z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,fn**\"&Ob'\"\"\"%\"kGF&%\"aG F&%\"gGF&F&**F%F&%\"fGF&F(F&F)F&!\"\"**\"&)G7F&%\"bGF&)F(\"\"#F&)F)F1F &F,**\"%#>)F&F/F&F(F&)F)\"\"$F&F&**\"%'4%F&%\"jGF&F2F&F(F&F&**F4F&F/F& )F(F6F&F)F&F&**F8F&F9F&F0F&F)F&F&*(F8F&F9F&F;F&F,*(\"$G\"F&)F(\"\"&F&F )F&F,*(\"%[?F&F/F&)F(\"\"%F&F,*(FCF&F/F&)F)FEF&F,*(F8F&%\"eGF&F;F&F,*( F.F&FIF&F5F&F&*(\"#kF&FDF&F2F&F,*(\"$c#F&F;F&F5F&F&*(FLF&F0F&FGF&F,*(F ?F&)F)FAF&F(F&F,*&FLF&)F(\"\"'F&F&*(\"&oF$F&F'F&F0F&F,*(F8F&F9F&F5F&F, *(FVF&F+F&F0F&F&*(FVF&F+F&F2F&F&*(FVF&F'F&F2F&F,**\"&![?F&FIF&F0F&F)F& F&**\"&s'GF&FIF&F2F&F(F&F,*(\"'s58F&FIF&F9F&F,*&FLF&)F)FTF&F&*&F%F&)FI F1F&F&*&F%F&)F9F1F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "so lve(%,k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*(\"$7&!\"\",V**\"%C5\" \"\"%\"fGF*%\"aGF*%\"gGF*F&**\"$#>F*%\"bGF*)F,\"\"#F*)F-F2F*F&**\"$G\" F*F0F*F,F*)F-\"\"$F*F***\"#kF*%\"jGF*F3F*F,F*F***F5F*F0F*)F,F7F*F-F*F* **F9F*F:F*F1F*F-F*F**(F9F*F:F*F " 0 "" {MPLTEXT 1 0 17 "A4:=eval(A3,k=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A4G,<*&),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-F,F-,&F)F-F .!\"\"F-F-*(%\"aGF-F+F-)F.\"\"$F-F-*&%\"bGF-)F.\"\"%F-F-*(,&F2F-%\"gGF -F-)F+F4F-)F.F,F-F-*(,,*&\"\")F0F;F,F0*(\"#;F0,2*&)F2F,F-F;F-F0*(\"\"& F-)F;F,F-F2F-F-*$)F2F4F-F0*&F4F-)F;F4F-F0*&\"#kF-%\"eGF-F-*&FOF-%\"jGF -F0*(FCF-F6F-F2F-F-*(FCF-F6F-F;F-F0F-,&F2F-F;F0F0F-F6F-*(F8F0F;F-F2F-F -*&FAF0F2F,F0F-F*F-F3F-F-*(FPF-F+F-F7F-F-*&%\"fGF-)F.FHF-F-*(F;F-)F+FH F-F.F-F-*,FCF0FDF-FUF0F+F8F.F,F-*(FRF-FF-F6F-FFF-FIF-F0**\"$G\"F-F6F-F2F-FMF-F-**FOF-FRF-F IF-F2F-F-**FboF-F6F-FKF-F;F-F-**FOF-FRF-FFF-F;F-F-*(FOF-FRF-FKF-F0*(F, F-)F2FHF-F;F-F0*(\"#KF-F6F-)F2F8F-F0*(FjoF-F6F-)F;F8F-F0*(FOF-FPF-FKF- F0*(F`oF-FPF-FMF-F-*&F[pF-FIF-F0*(F8F-FKF-FMF-F-*&FFF-F]pF-F0*(F,F-)F; FHF-F2F-F0*$)F2\"\"'F-F-*(FOF-FRF-FMF-F0*(F[oF-FZF-FFF-F-*(F[oF-FZF-FI F-F-**\"$?$F-FPF-FFF-F;F-F-**\"$[%F-FPF-FIF-F2F-F0*(\"%[?F-FPF-FRF-F0* $)F;FgpF-F-*&F^oF-)FPF,F-F-*&F^oF-)FRF,F-F-F-,(*(F,F-F;F-F2F-F0*$FIF-F -*$FFF-F-F0F+F,F.F8F-*(%\"lGF-F+F-FenF-F-*&%\"mGF-)F.FgpF-F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A4,x=0,y,0);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%-%'RootOfG6#,**&%\"mG\"\"\")%#_ZG\" \"$F*F**&%\"fGF*)F,\"\"#F*F*F*!\"\"*&%\"bGF*F,F*F**$)%\"xGF1F*,***\"5K K5>u9)[$*o$F2,bp**\"%[?F*F4F*)%\"gGF1F*%\"eGF*F***F=F*F4F*F>F*%\"jGF*F 2*,\"%'4%F*%\"aGF*F4F*F@F*F?F*F2**\"$G\"F*)FEF-F*F?F*FBF*F***\"$c#F*F> F*)FEF1F*F@F*F***F=F*F@F*FEF*FBF*F2**FGF*FEF*)F?F-F*FBF*F2**\"&S-\"F*F @F*F?F*FBF*F***\"$%QF*FHF*F?F*F@F*F2**FJF*F>F*FKF*FBF*F2**FRF*FEF*FNF* F@F*F**,FDF*FEF*F4F*FBF*F?F*F**(\"\"(F*)FE\"\"&F*F>F*F**&)FE\"\"%F*FNF 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1414 "discrim(-2752*g^5*a*e-8832*a^3*g^3*e-98304*e^2*j-12 992*a^5*e*g+14336*b*g^4*j-14336*b*g^4*e+27648*e^2*g^3+3072*f*a*g^4-11* g^8*a+1120*g^6*e+3072*f*a^4*g-32768*m*g^3-699*a^8*g-2048*f*a^2*g^3-204 8*f*a^3*g^2+1888*b*g^5*a^2-73728*b*g^2*a^2*e+73728*b*g^2*a^2*j-30720*g ^3*e*j-27648*e^2*a^2*g-96*g^7*a^2-18432*g^2*a*e*j+129024*a^2*g*j*e+186 6*g^4*a^5+27648*e^2*a^3+11040*b*g^3*a^4-7200*b*g^4*a^3-10240*b*a^4*e+1 0240*b*a^4*j+1696*g^2*a^7+98304*e*j^2-45056*b*a^3*j*g+424*g^6*a^3+3680 *e*a^6-1046*g^5*a^4-2264*g^3*a^6+224*b*g^6*a+19136*a^5*g*j-101376*a^2* g*j^2-160*b*g^7+3488*a^2*g^4*e-7328*a^2*g^4*j-864*j*g^6+3072*j^2*g^3+1 21*a^9+32768*e^3+9*g^9-32768*j^3+52224*j^2*a^3-4960*j*a^6-32768*f*g^2* e+32768*f*g^2*j-32768*f*a^2*e+32768*f*a^2*j-65536*f*a*g*j+65536*f*a*g* e+65536*b*e*j*g-32768*b*e^2*g-32768*b*j^2*g-65536*a*b*e*j-98304*a^2*m* g+98304*a*m*g^2+32768*a*b*e^2+32768*a*b*j^2+32768*a^3*m+53248*b*g^3*e* a-53248*b*g^3*j*a-27808*a^4*g^2*j+19072*a^3*g^3*j+3616*b*a^6*g+16288*a 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F1FQF&F&*(F8F&)F.F`oF&F1F&F2*(FSF&F-F&)FHF,F&F&*(FVF&F=F&)FHF8F&F&*(\" $O$F&FfoF&FIF&F2*(\"&OV\"F&F>F&FjoF&F&**\"&_\"\\F&F=F&FjoF&FIF&F2**\"% #>)F&F=F&FTF&F7F&F&**\"%/BF&F=F&FAF&FHF&F&**\"$'*)F&F=F&FAF&FIF&F2**\" &Wt&F&F=F&FjoF&F7F&F&**FbpF&F=F&FTF&FHF&F&**\"$7&F&FenF&F0F&FIF&F2**\" $%QF&F-F&F7F&FIF&F2**\"$c#F&F^oF&F1F&FIF&F&**\"%WhF&F5F&F1F&FTF&F2**\" $G\"F&FenF&F0F&FHF&F&**\"%'4%F&F-F&FIF&FHF&F2**\"%#z\"F&F-F&F7F&FHF&F& **\"$S'F&F:F&F>F&FHF&F&**FSF&F:F&F>F&FIF&F2**FcqF&F:F&F0F&FTF&F&**\"%[ ?F&F5F&FjoF&F1F&F&**F_qF&F^oF&FHF&F1F&F2**\"%SIF&F5F&F;F&FHF&F2**\"%g< F&F5F&F;F&FIF&F&**\"&Cm#F&F:F&F0F&FjoF&F2**\"&oF$F&F1F&F.F&F\\pF&F2**F hrF&F1F&F.F&FWF&F&**\"&g`\"F&F;F&F.F&FTF&F2*(\"$G&F&FfoF&FHF&F&*(FcqF& F>F&FTF&F&*(FVF&F0F&FWF&F2*(FVF&F0F&F\\pF&F&*,\"&c]%F&F=F&F7F&FHF&FIF& F2**F_rF&FjnF&F.F&FHF&F2**\"%_6F&FjnF&F.F&FIF&F&**\"&G\"[F&F;F&F.F&Fjo F&F2**FbpF&F0F&FjoF&FIF&F2**FbpF&F0F&FTF&FHF&F&**\"&K%=F&F>F&FHF&FIF&F 2*,FGF&F1F&F.F&FjoF&FIF&F&*,FGF&F1F&F.F&FTF&FHF&F2*,\"&[K&F&F;F&F.F&FH F&FIF&F&**FhpF&F+F&FAF&F:F&F&**\"%!G\"F&F+F&FjnF&F=F&F2**\"$s'F&F+F&F> F&F5F&F&**FhpF&F+F&F7F&FenF&F&*,\"&![?F&F+F&F7F&F:F&FIF&F2*,\"&?2$F&F+ F&F;F&F=F&FIF&F&*,FGF&%\"fGF&F=F&F0F&FHF&F2*,\"&Ob'F&F[uF&F.F&F7F&FHF& F&*,F]uF&F[uF&F:F&F1F&FHF&F&*,FcqF&F+F&F-F&FIF&F1F&F&*(FVF&%\"mGF&F>F& F2*(FVF&F-F&FauF&F&*(F]qF&F[uF&FjnF&F2**F_rF&F+F&FenF&FIF&F2**\"'SQ;F& FauF&F7F&F=F&F2**\"&?>)F&FauF&F;F&F.F&F&**FfuF&FauF&F0F&F:F&F&**FhuF&F 5F&FauF&F1F&F2**\"%3YF&F[uF&F>F&F=F&F2**\"%gDF&F[uF&FAF&F.F&F&**F^vF&F [uF&F;F&F:F&F&**FcqF&F+F&FAF&FHF&F2**FgqF&F+F&FAF&FIF&F&**F^vF&F[uF&F5 F&F7F&F&**\"%o:F&F+F&F;F&F-F&F2*,\"&S9'F&F+F&F;F&FHF&F=F&F2*,FitF&F+F& F>F&FHF&F.F&F&*,FfvF&F+F&F7F&FHF&F:F&F&*,F[tF&F+F&F>F&FIF&F.F&F2*,FcqF &F+F&F-F&FHF&F1F&F&*,FitF&F+F&F5F&FHF&F0F&F2**\"$C'F&F+F&FfoF&F.F&F&** \"$w\"F&F+F&F1F&FPF&F2**F\\vF&F[uF&F-F&F0F&F2**F^vF&F[uF&FenF&F1F&F&*( F]qF&F[uF&F^oF&F2*(\"#[F&F+F&FaoF&F&*,F]uF&F[uF&F7F&FIF&F.F&F2*,FGF&F[ uF&F0F&FIF&F=F&F&*,F]uF&F[uF&F:F&FIF&F1F&F2*(\"$7\"F&F+F&FLF&F2**FVF&F [uF&F;F&FHF&F2**FVF&F[uF&F;F&FIF&F&**FVF&F[uF&F5F&FHF&F2**FVF&F[uF&F5F &FIF&F&**FVF&F+F&F7F&FjoF&F2**FVF&F+F&F7F&FTF&F2*,FbpF&F+F&F0F&FjoF&F. F&F&*,FbpF&F+F&F0F&FTF&F.F&F&*,FhrF&F+F&F7F&FHF&FIF&F&**FVF&F+F&F:F&Fj oF&F&**FVF&F+F&F:F&FTF&F&*,FbpF&F+F&F=F&FjoF&F1F&F2*,FbpF&F+F&F=F&FTF& F1F&F2*,FhrF&F+F&F:F&FHF&FIF&F2*.FGF&F+F&F=F&FHF&FIF&F1F&F&*(F]qF&F*F& F^oF&F2*(F]qF&F*F&FjnF&F&**F@F&F*F&FenF&F1F&F&*(\"#AF&)F1F`oF&F.F&F2*, F`pF&F5F&F1F&FIF&FHF&F&F&,&F.F&F1F2F&F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*(\"' )GC&\"\"\"),&%\"aGF&%\"gG!\"\"\"\"$F&,du**\"$#**F&)F*\"\"&F&F)F&%\"eGF &F+**\"%;7F&)F)F,F&)F*F,F&F2F&F&*(\"&_\"\\F&)F2\"\"#F&%\"jGF&F+**\"$C# F&)F)F1F&F2F&F*F&F+**\"%'4%F&%\"bGF&)F*\"\"%F&F;F&F&**\"%WhF&FAF&FBF&F 2F&F+*(\"&OV\"F&F9F&F6F&F&**\"%O:F&%\"fGF&F)F&FBF&F&*(\"#7F&)F*\"\")F& F)F&F+*(\"$G&F&)F*\"\"'F&F2F&F&**FIF&FJF&)F)FCF&F*F&F&*(\"&%Q;F&%\"mGF &F6F&F+*(F,F&)F)FNF&F*F&F+**\"%C5F&FJF&)F)F:F&F6F&F+**FenF&FJF&F5F&)F* F:F&F+**\"$o$F&FAF&F0F&FfnF&F+*(\"$7&F&F>F&)FAF:F&F+*,\"&K%=F&FAF&FhnF &FfnF&F2F&F+*,FEF&FAF&FhnF&FfnF&F;F&F&**F_oF&F6F&F2F&F;F&F+**F@F&F9F&F fnF&F*F&F&*(FCF&)F*\"\"(F&FfnF&F+*,FVF&FhnF&F)F&F2F&F;F&F&*,FEF&FfnF&F *F&F;F&F2F&F&*(\"#?F&FBF&F>F&F+*(FenF&F9F&F5F&F&**\"$g&F&FAF&F6F&FTF&F &**\"$S#F&FAF&FBF&F5F&F+**\"%[?F&FAF&FTF&F;F&F+*(FCF&FhnF&)F)FeoF&F&*( F\\oF&F]oF&F0F&F&*(F8F&F2F&)F;F:F&F&**\"%gDF&F)F&F]oF&FBF&F+*,F`pF&FAF &F5F&F;F&F*F&F&*(\"#GF&FQF&F5F&F&*(\"#!)F&F2F&)F)FRF&F&*(\"#5F&F0F&FTF &F+*(FLF&F6F&F]qF&F&**\"$+%F&FAF&FQF&F)F&F&**F=F&F>F&F*F&F;F&F&**F@F&F fnF&F*F&FepF&F+*(\"$7\"F&FAF&FdoF&F+**\"$3#F&FfnF&FBF&F2F&F+**FbqF&Ffn F&FBF&F;F&F&*(\"$O$F&F;F&FQF&F+*(FEF&FepF&F6F&F&*&FVF&)F2F,F&F&*&F1F&) F*\"\"*F&F&*&FVF&)F;F,F&F+*(FenF&FepF&F5F&F&*(\"#;F&F;F&F]qF&F+**FVF&F JF&FhnF&F2F&F+**FVF&FJF&FhnF&F;F&F&**FVF&FJF&FfnF&F2F&F+**FVF&FJF&FfnF &F;F&F&*,\"&oF$F&FJF&F)F&F*F&F;F&F+*,F\\sF&FJF&F)F&F*F&F2F&F&*,F\\sF&F AF&F2F&F;F&F*F&F&**FVF&FAF&F9F&F*F&F+**FVF&FAF&FepF&F*F&F+*,F\\sF&F)F& FAF&F2F&F;F&F+**F8F&FfnF&FWF&F*F&F+**F8F&F)F&FWF&FhnF&F&**FVF&F)F&FAF& F9F&F&**FVF&F)F&FAF&FepF&F&*(FVF&F5F&FWF&F&**FgpF&FTF&F]oF&F*F&F&**\"% ?^F&F5F&F]oF&FhnF&F+**FisF&FfnF&F]oF&F6F&F&*,F_oF&FAF&F6F&F2F&F)F&F&*, \"&S-\"F&FAF&F6F&F;F&F)F&F+**\"#[F&FTF&FhnF&F;F&F+**\"$/(F&F5F&F6F&F;F &F+**F\\qF&FAF&F]qF&F*F&F+**FbqF&FTF&FhnF&F2F&F+**F@F&F5F&F;F&F2F&F+*( F\\oF&FJF&F>F&F+*(F\\oF&FJF&F0F&F+**FhqF&FAF&FhnF&F>F&F+*,FEF&FAF&F5F& F2F&F*F&F&**\"&c%>F&FhnF&F)F&F9F&F+**\"$![F&F0F&F)F&F;F&F&**\"%sIF&Fhn F&F)F&FepF&F+*(F_tF&FAF&FbpF&F&F&F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Because the discriminant factors, we have two cases to consider . Case B is the shorter factor. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B:=eval(A5,a=g);" }}{PARA 8 "" 1 "" {TEXT -1 43 "Erro r, numeric exception: division by zero\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Back to case A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1101 "solve(-992*g^5*a*e+1216*a^3*g^3*e-49152*e^2*j-224*a^5*e*g+4096*b *g^4*j-6144*b*g^4*e+14336*e^2*g^3+1536*f*a*g^4-12*g^8*a+528*g^6*e+1536 *f*a^4*g-16384*m*g^3-3*a^8*g-1024*f*a^2*g^3-1024*f*a^3*g^2-368*b*g^5*a ^2-512*a^5*b^2-18432*b*g^2*a^2*e+6144*b*g^2*a^2*j-18432*g^3*e*j+4096*e ^2*a^2*g-4*g^7*a^2+16384*g^2*a*e*j+6144*a^2*g*j*e-20*g^4*a^5+1024*e^2* a^3+560*b*g^3*a^4-240*b*g^4*a^3-2048*b*a^4*j+4*g^2*a^7+512*b^2*g^5+491 52*e*j^2-2560*a*b^2*g^4+2048*b*a^3*j*g+28*g^6*a^3+80*e*a^6-10*g^5*a^4+ 12*g^3*a^6+400*b*g^6*a+224*a^5*g*j-4096*a^2*g*j^2-112*b*g^7-208*a^2*g^ 4*e+400*a^2*g^4*j-336*j*g^6+6144*j^2*g^3+16384*e^3+5*g^9-16384*j^3+102 4*j^2*a^3-16*j*a^6-16384*f*g^2*e+16384*f*g^2*j-16384*f*a^2*e+16384*f*a ^2*j-32768*f*a*g*j+32768*f*a*g*e+32768*b*e*j*g-16384*b*e^2*g-16384*b*j ^2*g-32768*a*b*e*j-49152*a^2*m*g+49152*a*m*g^2+16384*a*b*e^2+16384*a*b *j^2+16384*a^3*m+2560*a^4*b^2*g-5120*a^3*b^2*g^2+5120*a^2*b^2*g^3+1843 2*b*g^3*e*a-10240*b*g^3*j*a-48*a^4*g^2*j-704*a^3*g^3*j-80*b*a^6*g-400* a^4*g^2*e-4096*a^3*j*e-512*f*a^5-512*f*g^5-208*b*g^2*a^5+6144*b*a^3*e* g-19456*g^2*a*e^2+480*g^5*a*j-3072*g^2*a*j^2+48*b*a^7,m);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*(\"&%Q;!\"\",\\u**\"$#**\"\"\")%\"gG\"\"&F*% \"aGF*%\"eGF*F&**\"%;7F*)F.\"\"$F*)F,F3F*F/F*F**(\"&_\"\\F*)F/\"\"#F*% \"jGF*F&**\"$C#F*)F.F-F*F/F*F,F*F&**\"%'4%F*%\"bGF*)F,\"\"%F*F9F*F*** \"%WhF*F?F*F@F*F/F*F&*(\"&OV\"F*F7F*F4F*F***\"%O:F*%\"fGF*F.F*F@F*F**( \"#7F*)F,\"\")F*F.F*F&*(\"$G&F*)F,\"\"'F*F/F*F***FGF*FHF*)F.FAF*F,F*F* *(F3F*)F.FLF*F,F*F&**\"%C5F*FHF*)F.F8F*F4F*F&**FVF*FHF*F2F*)F,F8F*F&** \"$o$F*F?F*F+F*FWF*F&*(\"$7&F*FF*F7F*FWF*F,F*F**(FA F*)F,\"\"(F*FWF*F&*,F%F*FYF*F.F*F/F*F9F*F**,FCF*FWF*F,F*F9F*F/F*F**(\" #?F*F@F*FF*FWF*F,F*F`pF*F&*(\"$7 \"F*F?F*F_oF*F&**\"$3#F*FWF*F@F*F/F*F&**F]qF*FWF*F@F*F9F*F**(\"$O$F*F9 F*FOF*F&*(FCF*F`pF*F4F*F**&F%F*)F/F3F*F**&F-F*)F,\"\"*F*F**&F%F*)F9F3F *F&*(FVF*F`pF*F2F*F**(\"#;F*F9F*FhpF*F&**F%F*FHF*FYF*F/F*F&**F%F*FHF*F YF*F9F*F***F%F*FHF*FWF*F/F*F&**F%F*FHF*FWF*F9F*F**,\"&oF$F*FHF*F.F*F,F *F9F*F&*,FgrF*FHF*F.F*F,F*F/F*F**,FgrF*F?F*F/F*F9F*F,F*F***F%F*F?F*F7F *F,F*F&**F%F*F?F*F`pF*F,F*F&*,FgrF*F.F*F?F*F/F*F9F*F&**F%F*F.F*F?F*F7F *F***F%F*F.F*F?F*F`pF*F***FbpF*FRF*FhnF*F,F*F***\"%?^F*F2F*FhnF*FYF*F& **FasF*FWF*FhnF*F4F*F**,FjnF*F?F*F4F*F/F*F.F*F**,\"&S-\"F*F?F*F4F*F9F* F.F*F&**\"#[F*FRF*FYF*F9F*F&**\"$/(F*F2F*F4F*F9F*F&**FgpF*F?F*FhpF*F,F *F&**F]qF*FRF*FYF*F/F*F&**F>F*F2F*F9F*F/F*F&*(FgnF*FHF*FF*FYF*F.F *F7F*F&**\"$![F*F+F*F.F*F9F*F***\"%sIF*FYF*F.F*F`pF*F&*(FgsF*F?F*F]pF* F*F*,**(F3F*FWF*F,F*F&*(F3F*FYF*F.F*F**$F2F*F**$F4F*F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A6:=eval(A5,m=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 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-F0*,FfzF-FZF-F2F-F;F-FPF-F-*,FfzF-F6F-FPF-FRF-F;F-F-**F`uF-F6F-FdqF-F ;F-F0**F`uF-F6F-FfqF-F;F-F0*,FfzF-F2F-F6F-FPF-FRF-F0**F`uF-F2F-F6F-Fdq F-F-**F`uF-F2F-F6F-FfqF-F-**FcxF-F[pF-F^wF-F;F-F-**\"%?^F-FKF-F^wF-FIF -F0**F`[lF-FFF-F^wF-FMF-F-*,F`wF-F6F-FMF-FPF-F2F-F-*,FgrF-F6F-FMF-FRF- F2F-F0**\"#[F-F[pF-FIF-FRF-F0**\"$/(F-FKF-FMF-FRF-F0**FhxF-F6F-FfpF-F; F-F0**F]yF-F[pF-FIF-FPF-F0**F\\rF-FKF-FRF-FPF-F0*(F[oF-FZF-FhoF-F0*(F[ oF-FZF-FdpF-F0**FcyF-F6F-FIF-FhoF-F0*,F\\vF-F6F-FKF-FPF-F;F-F-**\"&c%> F-FIF-F2F-FdqF-F0**\"$![F-FdpF-F2F-FRF-F-**FjsF-FIF-F2F-FfqF-F0*(Fe[lF -F6F-FbtF-F-F-,**(F4F-FFF-F;F-F0*(F4F-FIF-F2F-F-FJF-*$FMF-F0F0F.FgpF0 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "reevaluate polynomial [2]" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A6,x=0,y,0);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%-%'RootOfG6#,0*&,\\u**\"$#**\"\"\")% \"gG\"\"&F,%\"aGF,%\"eGF,!\"\"**\"%;7F,)F0\"\"$F,)F.F6F,F1F,F,*(\"&_\" \\F,)F1\"\"#F,%\"jGF,F2**\"$C#F,)F0F/F,F1F,F.F,F2**\"%'4%F,%\"bGF,)F. \"\"%F,FF,F?F,F.F,FF,FfnF,F0F,F:F,F2**\"$![F,F-F,F0F,F< F,F,**\"%sIF,FfnF,F0F,FdpF,F2*(F[tF,FBF,FapF,F,F,)%#_ZGF6F,F,*&,***F9F ,FKF,F0F,FfnF,F2*(FeoF,FKF,F5F,F2**F9F,FKF,FZF,F.F,F,*(FeoF,FKF,F7F,F, F,)F]uF;F,F,*&,**(FeoF,FBF,F5F,F2*(FeoF,FBF,F7F,F,**F9F,FBF,FZF,F.F,F, **F9F,FBF,F0F,FfnF,F2F,F]uF,F,*(F9F,FZF,F.F,F2*&FeoF,F7F,F2*&FeoF,F5F, F,*(F9F,FfnF,F0F,F,*$)%\"xGF;F,,,*,\"'W@EF2,^y*&\"#=F,)F.F^qF,F2*$)F0F ^qF,F2**\"&+c#F,F:F,F5F,F.F,F2**FftF,F:F,FZF,FfnF,F,**\"&kC'F,F:F,F7F, F0F,F,**FesF,FdpF,F5F,F.F,F2**\"&[w#F,FdpF,FZF,FfnF,F,**\"%;#*F,FdpF,F 7F,F0F,F,**\"%!3#F,FBF,FCF,FUF,F2**FJF,FBF,F-F,F5F,F,**\"$G\"F,FBF,F7F ,F?F,F,**FhnF,F1F,F7F,FUF,F,**\"%w\\F,F1F,FCF,F5F,F2**FAF,F1F,F)F,FKF,FfnF,FF,F1F,FboF,F2*(F[qF,F1F,FapF,F2*(\"#KF,FapF,F7F,F2*(\"#MF,F\\qF,FCF ,F2*(\"#%*F,F?F,F-F,F,*(FMF,FUF,FRF,F,*(\"$/\"F,F5F,FboF,F2*(F_pF,F:F, FUF,F,*(\"&o$eF,F:F,FCF,F2*(FAF,FdpF,FUF,F2*(FbwF,FdpF,FCF,F2*,\"&?2$F ,F1F,FF,F " 0 "" {MPLTEXT 1 0 1630 "factor(-18*g^10-a^10-25600*e^2*a^3*g+19456*e^2*a^2*g^2+62464 *e^2*g^3*a-5120*j^2*a^3*g+27648*j^2*a^2*g^2+9216*j^2*g^3*a-2080*b*g^4* a^4+1536*b*g^5*a^3+128*b*g^3*a^5+368*e*g^3*a^4-4976*e*g^4*a^3+4096*e*j *a^4+79872*e*j*g^4-16384*f*g^4*b-80*j*a^6*g-1296*j*a^5*g^2+4096*b*a^5* e+944*j*g^3*a^4+2768*j*g^4*a^3-2160*j*g^5*a^2+512*f*a^6-24576*b*a^3*j* g^2-360448*b*e*j*g^2-98304*a^2*b*e*j-180224*f*g^2*e*a+81920*f*g^2*j*a+ 114688*f*a^2*e*g-16384*f*a^2*j*g-294912*b*e^2*g*a-163840*b*j^2*g*a+153 6*f*g^6+3*a^9*g+14*a^8*g^2+112*j*a^7+27*g^8*a^2+39*g^9*a+1296*j*g^7+43 2*b*g^8-16*b*a^8-1968*e*g^7-80*e*a^7-32*a^7*g^3-34*a^6*g^4+94*a^5*g^5+ 12*a^4*g^6-104*a^3*g^7+2048*e^2*a^4-58368*e^2*g^4-4096*j^2*a^4-27648*j ^2*g^4+30720*e*j*g*a^3-59392*e*j*g^2*a^2-55296*e*j*g^3*a-16384*a^4*f*b -17920*a^2*b^2*g^4-1584*j*g^6*a+896*b*g^6*a^2-1408*b*g^7*a-256*b*a^7*g +768*b*a^6*g^2+2000*e*g^5*a^2+3216*e*g^6*a-400*e*a^6*g+1840*e*a^5*g^2+ 49152*e^3*a-245760*e^3*g+49152*j^3*a+147456*j^3*g+81920*f*g^3*e-49152* f*g^3*j-16384*f*a^3*e-16384*f*a^3*j+212992*b*e^2*g^2+147456*b*j^2*g^2+ 81920*a^2*b*e^2+16384*a^2*b*j^2+2560*a^4*b^2*g^2+10240*a^3*b^2*g^3-491 52*e^2*j*a+638976*e^2*j*g-5120*a^5*b^2*g+11264*b^2*g^5*a-49152*e*j^2*a -540672*e*j^2*g+1536*f*g^4*a^2-4096*f*g^5*a+4096*f*g^3*a^3-3584*f*a^4* g^2+26624*b*g^5*e-18432*b*g^5*j+458752*b*e*j*g*a+45056*b*g^3*e*a^2-696 32*b*g^4*e*a-12288*b*g^3*j*a^2+40960*b*g^4*j*a-22528*b*a^4*e*g+16384*b *a^3*e*g^2+14336*b*a^4*j*g-32768*b^2*g^3*e+32768*b^2*g^3*j+98304*a*b^2 *e*g^2-98304*a*b^2*j*g^2+65536*a*f*b*g^3+65536*a^3*f*b*g+98304*a^2*b^2 *j*g-98304*a^2*b^2*e*g-98304*a^2*f*b*g^2+32768*a^3*b^2*e-32768*a^3*b^2 *j+1536*a^6*b^2-2560*b^2*g^6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,: *(\"#K\"\"\"%\"eGF'%\"aGF'F'*(\"$g\"F'F(F'%\"gGF'!\"\"*(\"\"#F')F)\"\" $F'F,F'F-*(\"\"%F')F,F/F')F)F/F'F'*(F/F'F)F')F,F1F'F'*$)F)F3F'F-*&F1F' )F,F3F'F-*(F&F'%\"jGF'F)F'F'*(\"#'*F'F=F'F,F'F'**\"#kF'%\"bGF'F,F'F)F' F-*(F&F'FBF'F4F'F'*(F&F'FBF'F5F'F'F',fn**\"%C5F'FBF'F=F'F,F'F'**FGF'FB F'F(F'F,F'F-**FGF'%\"fGF'F)F'F,F'F'**F?F'FBF'F5F'F4F'F-**\"$#>F'FBF'F) F'F7F'F'**\"$W\"F'F=F'F4F'F)F'F'**FAF'FBF'F0F'F,F'F-**\"$w\"F'F=F'F5F' F,F'F'*(\"#!)F'F=F'F0F'F-*(\"\"&F')F)FVF'F,F'F-*(\"#[F'FBF'F9F'F'*(FTF 'FBF'F;F'F-*(\"$7\"F'F(F'F0F'F'*(\"$O$F'F(F'F7F'F'*(\"#9F'F0F'F7F'F'*( \"\"(F'F5F'F;F'F-*(\"\"*F')F,FVF'F)F'F-*$)F)\"\"'F'F'*(\"$S#F'F=F'F7F' F-*(\"$7&F'FJF'F5F'F-*(FfoF'FJF'F4F'F-**FOF'F(F'F5F'F,F'F-**\"$/$F'F(F 'F4F'F)F'F-*(\"%sIF'F(F'F=F'F-*&FboF')F,FboF'F'*&\"%O:F')F(F/F'F'*&F`p F')F=F/F'F'**FGF'F)F'F=F'FBF'F-**FGF'F(F'FBF'F)F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Case C: short factor=0; Case A: long fact or=0 [7]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "solve(32*e*a-1 60*e*g-2*a^3*g+4*g^2*a^2+2*a*g^3-a^4-3*g^4+32*j*a+96*j*g-64*b*g*a+32*b *g^2+32*b*a^2,j);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#K!\"\",6*( F%\"\"\"%\"eGF)%\"aGF)F)*(\"$g\"F)F*F)%\"gGF)F&*(\"\"#F))F+\"\"$F)F.F) F&*(\"\"%F))F.F0F))F+F0F)F)*(F0F)F+F))F.F2F)F)*$)F+F4F)F&*&F2F))F.F4F) F&*(F%F)%\"bGF)F6F)F)**\"#kF)F>F)F.F)F+F)F&*(F%F)F>F)F5F)F)F),&F+F)*&F 2F)F.F)F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "C:=eval(A6 ,j=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"CG,<*&),&*$)%\"xG\"\"#\" \"\"F-%\"yGF-F,F-,&F)F-F.!\"\"F-F-*(%\"aGF-F+F-)F.\"\"$F-F-*&%\"bGF-)F .\"\"%F-F-*(,&F2F-%\"gGF-F-)F+F4F-)F.F,F-F-*(,,*&\"\")F0F;F,F0*(\"#;F0 ,2*&)F2F,F-F;F-F0*(\"\"&F-)F;F,F-F2F-F-*$)F2F4F-F0*&F4F-)F;F4F-F0*&\"# kF-%\"eGF-F-*(F,F-,6*(\"#KF-FPF-F2F-F-*(\"$g\"F-FPF-F;F-F0*(F,F-FKF-F; F-F0*(F8F-FIF-FFF-F-*(F,F-F2F-FMF-F-*$)F2F8F-F0*&F4F-)F;F8F-F0*(FTF-F6 F-FFF-F-**FOF-F6F-F;F-F2F-F0*(FTF-F6F-FIF-F-F-,&F2F-*&F4F-F;F-F-F0F-*( FCF-F6F-F2F-F-*(FCF-F6F-F;F-F0F-,&F2F-F;F0F0F-F6F-*(F8F0F;F-F2F-F-*&FA F0F2F,F0F-F*F-F3F-F-*(FPF-F+F-F7F-F-*&%\"fGF-)F.FHF-F-*(F;F-)F+FHF-F.F -F-*,FCF0FDF-F_oF0F+F8F.F,F-*,FTF0FRF-F[oF0F+F4F.F4F0*,\"$7&F0,V**\"%C 5F-FdoF-F2F-F;F-F0**\"$#>F-F6F-FFF-FIF-F0**\"$G\"F-F6F-F2F-FMF-F-**Fbp F-F6F-FKF-F;F-F-*(F,F-)F2FHF-F;F-F0*(FTF-F6F-FenF-F0*(FTF-F6F-FgnF-F0* 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F-F[oF0FPF-F-*(F[pF-FdoF-FepF-F0*(F[pF-FdoF-F^qF-F0*.FeqF-F6F-F;F4FRF- F[oF0F2F-F-*,\"#:F-F;FHF2F-FRF-F[oF0F0*,F4F-F;F,F2F-FRF,F[oF`rF0**FazF -F6F-FIF-FepF-F0*,FevF-F6F-FKF-FPF-F;F-F-**\"&c%>F-FIF-F2F-F[rF-F0*(Fj yF-F6F-FctF-F-F-,**(F4F-FFF-F;F-F0*(F4F-FIF-F2F-F-FJF-*$FMF-F0F0F.FaqF 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [2]" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(C);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**\"#K!\"\",6*(\"#7\"\"\"%\"gGF*%\"yGF*F**()%\"a G\"\"#F*%\"xGF*F,F*F&**\"\"$F*F1F*)F+F0F*F,F*F**(\"\")F*)F,F0F*%\"eGF* F***\"\"%F*F/F*F7F*%\"bGF*F**(F:F*F/F*F,F*F**(F:F*F/F*)F1F0F*F**(F)F*F >F*F+F*F**,F0F*F/F*F1F*F+F*F,F*F&**F:F*F7F*F+F*F;F*F&F*,`p*,\"\"'F*F>F *F7F*)F/F3F*F+F*F***\"#sF*)F,F:F*%\"fGF*F4F*F&**F6F*FHF*FIF*F.F*F&**\" #[F*F8F*)F,F3F*F+F*F&**\"#;F*F8F*FMF*F/F*F&*(F%F*FHF*)F8F0F*F***FLF*F/ F*)F1F:F*F+F*F&*.FOF*F8F*FMF*F/F*F1F*F+F*F&*.\"#'*F*F7F*F/F*F>F*F;F*F+ F*F&*,FLF*F8F*FHF*F;F*F/F*F**,FOF*F8F*F;F*FHF*F+F*F**,FLF*F8F*F1F*FMF* F4F*F&*,FLF*F8F*F7F*F/F*F>F*F&*,\"$W\"F*F8F*F>F*F7F*F+F*F&*(F>F*F7F*)F /F:F*F***\"\"*F*F>F*F7F*)F+F:F*F&*,F6F*F>F*F7F*F.F*F4F*F**,FDF*F>F*F7F *F/F*)F+F3F*F&*,FDF*FMF*F1F*F;F*F^oF*F**,F0F*FMF*FEF*F1F*F;F*F**,\"#kF *FMF*F;F*F+F*F/F*F&*,\"#QF*F7F*F1F*F.F*F+F*F&*,F%F*F7F*F.F*F>F*F;F*F&* *FOF*FHF*)F;F0F*F4F*F&**FOF*FHF*F.F*FgoF*F***FLF*FMF*F;F*F4F*F&**FOF*F MF*F;F*F.F*F&**FDF*F7F*F1F*FEF*F&**\"#=F*F7F*F1F*F^oF*F&**\"#aF*F,F*)F 1F3F*F^oF*F&**F0F*F,F*F`pF*FEF*F&*(FGF*F7F*F4F*F**(F6F*F.F*F7F*F***FLF *F/F*F7F*F+F*F**,FLF*FHF*FIF*F/F*F+F*F&*.\"#5F*FMF*F/F*F1F*F;F*F4F*F&* .F0F*FMF*F.F*F1F*F;F*F+F*F**,\"#mF*F7F*F1F*F4F*F/F*F&*,F]pF*F,F*F`pF*F .F*F+F*F&*,F_pF*F,F*F`pF*F4F*F/F*F&*(F6F*F.F*FSF*F&*(FGF*FSF*F4F*F&F*, &F/F**&F3F*F+F*F*!\"$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "C fact ors, so it is complete [5]. Back to case A. 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[g%*)p^9F&F+F&FOF&F?F&F0F&F&*,\".![sv3K;F&F+F&F0F&FhnF&F*F&F&*,\".wxi6 &*4\"F&F+F&F0F&FRF&F;F&F,**\",wP!)pd#F&F/F&FhnF&F*F&F&**\"-)o-W_g\"F&F 0F&F]pF&FBF&F&*,\".sux!=67F&F0F&FhnF&FeoF&F;F&F,**\"/g\\i6!RZ#F&FhoF&F boF&FapF&F,**F[\\lF&F2F&FVF&FapF&F&**\",_.Qh\\'F&Fj_lF&F0F&FeoF&F,**\" -g4SnrwZ-&F&F`_lF&F8F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**\"( /V>%\"\"\"),&%\"aGF&%\"gG!\"\"\"\"&F&,2*(\"#F& F)F&F&*(\"$g\"F&F>F&F*F&F+*(F1F&F7F&F*F&F+*(\"\"%F&F4F&F0F&F&*(F1F&F)F &F;F&F&*$)F)FMF&F+*&F8F&)F*FMF&F+*(FHF&FCF&F)F&F&*(\"#'*F&FCF&F*F&F&** \"#kF&FAF&F*F&F)F&F+*(FHF&FAF&F4F&F&*(FHF&FAF&F0F&F&F1F&F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Note that factor (a-g) has already been \+ considered. The next shortest factor is considered in case D. The long est is considered in case A. [7]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "solve(-17*a^2*g-15*g^2*a+11*a^3+21*g^3+384*e+128*b*a- 384*j-128*b*g,j);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0**\"#<\"\"\"\"$ %Q!\"\"%\"aG\"\"#%\"gGF&F(**\"\"&F&\"$G\"F(F+F*F)F&F(*(\"#6F&F'F(F)\" \"$F&*(\"\"(F&F.F(F+F1F&%\"eGF&*(F1F(%\"bGF&F)F&F&*(F1F(F6F&F+F&F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "D1:=eval(A7,j=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#D1G,<*&),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-F,F -,&F)F-F.!\"\"F-F-*(%\"aGF-F+F-)F.\"\"$F-F-*&%\"bGF-)F.\"\"%F-F-*(,&F2 F-%\"gGF-F-)F+F4F-)F.F,F-F-*(,,*&\"\")F0F;F,F0*(\"#;F0,.*&#\"#6\"\"'F- *&)F2F,F-F;F-F-F-*&#\"#:F,F-*&)F;F,F-F2F-F-F-*&#\"#F-F6F-F2F-FYF-F-**\"#kF-F6F-FTF-F;F-F0*(\"\"&F-)F2F[pF-F;F-F0*(\"#[ F-F6F-)F2F8F-F-*(\"#!)F-F6F-)F;F8F-F0*(\"$7\"F-F^oF-FTF-F-*(\"$O$F-F^o F-FYF-F-*(\"#9F-FTF-FYF-F-*(\"\"(F-FJF-FbpF-F0*(\"\"*F-)F;F[pF-F2F-F0* $)F2FHF-F-**\"$W\"F-F^oF-FJF-F;F-F0**\"$/$F-F^oF-FOF-F2F-F0*&FHF-)F;FH F-F-*&\"%O:F-)F^oF,F-F-*(\"%sIF-F^oF-,0**FRF-\"$%QF0F2F,F;F-F0**F[pF- \"$G\"F0F;F,F2F-F0*(FGF-F]rF0F2F4F-*(FjpF-F_rF0F;F4F-F^oF-*(F4F0F6F-F2 F-F-*(F4F0F6F-F;F-F0F-F0**FcoF-F6F-F[rF-F;F-F-*&FgqF-)F[rF,F-F-*(\"$S# F-F[rF-FYF-F0**\"$w\"F-F[rF-FJF-F;F-F-**FcoF-F2F-F[rF-F6F-F0**FcoF-F^o F-F6F-F2F-F-**FaqF-F[rF-FOF-F2F-F-*(FapF-F[rF-FTF-F0F-,(*(F,F-F;F-F2F- F0*$FOF-F-*$FJF-F-F0F.F[pF-*(F;F-)F+F[pF-F.F-F-*,FCF0FDF-FjnF0F+F8F.F, F-*(F[rF-FF-FOF-F2F-FhqF-F0*(F^pF-F6F-F ]wF-F-F-,**(F4F-FJF-F;F-F0*(F4F-FOF-F2F-F-FSF-FXF0F0F.FHF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(D1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,`u*$)%\"xG\"\"'\"\"\"F(*0\"\"$F(\"#k!\"\"%\"aGF(F&F(% \"yG\"\"&%\"bGF(%\"gG\"\"#F(*(F-F(F&F()F.F*F(F(*(%\"eGF(F&F()F.\"\"%F( F(*(F1F()F&F/F(F.F(F(*&F0F(F7F(F(*$F4F(F,*&)F&F8F(F.F(F(*&)F&F2F()F.F2 F(F,*,\"#FF(\"&%Q;F,F&F(F.F/F1F/F,*,\"\"*F(\"%[?F,F.F'F6F(F1F*F,*,FCF( FDF,F.F'F0F(F1F8F(*,\"#8F(\"#KF,F&F8F.F2F1F2F(*.\"#)F,F-F2F&F(F.F/F1F*F(*. 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FFF.F0F.F)**F-F.F/F.FOF.FFF.F)*(\"$S&F.FEF.F0F.F.*(\"%3YF.F/F.FEF.F)*( FHF.FLF.FIF.F.*(\"%:7F.FLF.F6F.F.*(\"%e9F.)F1\"\"&F.FFF.F.*(\"#=F.)FFF hnF.F1F.F.*$)FFF;F.F.**\"(s%pF.FEF.F.*(F_oF.FAF.FEF.F)**\"'S%H)F.F5F.F0F.Fbo F.F.**\"(Si'=F.F5F.F6F.FEF.F.**\"('HfBF.F5F.FboF.F/F.F)*(\"$C$F.FOF.)F FF]sF.F)*(\"&O7'F.F:F.FEF.F)*(\"%/BF.F/F.F]oF.F.*(\"&=1$F.FgnF.FboF.F) *(\"%oAF.F0F.F]oF.F)**\"(W(>6F.F5F.F:F.FFF.F.**\"(g$*z#F.FLF.F6F.F/F.F .*$)FF\"\"*F.F)*&\"*K[)HXF.)F/F2F.F.*&\"&$o>F.)F1FjtF.F)**\"*k'pf!*F.F FF.F5F.FAF.F.**\"())y2(F.FboF.FIF.F1F.F)**\")kOB@F.FEF.FIF.FOF.F)**Ffu F.FLF.FIF.F0F.F)*,FgrF.F5F.F0F.F/F.FFF.F)**\"&[w#F.F5F.F]oF.F1F.F.**\" 'S5JF.FboF.FOF.F/F.F.**\"*w(zRgF.F/F.FLF.FIF.F.**\"'gt?F.F5F.FOF.F[oF. F.*,FfuF.F5F.FEF.F/F.F1F.F)**FhqF.FOF.FFF.FAF.F)*(\"%O:F.F5F.F[tF.F.F] s*&F]qF.,\\p*,\"$V#F.\"(/V>%F)F1FhnFFF.F/F.F.*,\"#XF.\"(_r4#F)FFF2F1F2 F/F.F.*,F2F.FivF)FFFhnF/F.F1F.F.**F`rF.\"&oF$F)F/FBF1F2F)**\"%(=#F.\"+ [O[Z@F)F1F\\rFFF.F)**FhvF.\"(3')Q)F)F1F;F/F.F.*(FbwF)FFF\\rF1F.F)*,\"# \")F.F\\wF)F5F.F1FhnFFFBF.*(\"&GP(F)FFFhnF5FBF)*.FjtF.\"%#>)F)F5F.F1FB FFFBF/F.F)*,FjtF.F_wF)F/FBFFFBF1F.F)**F9F.\"*74(o`F)F1F]sFFFBF)**\"$*= F.\"+C=ut5F)F1F7FFFhnF)*(\"$K%F)F5F2FFF2F.*(F_wF)F/FBFFF2F)*,\"#:F.F=F )F5F.F1F2FFF7F.*,FdoF.FivF)F5F.F1F7FFF2F.**\"&wX#F)F5F.FFF7F/F.F)**F2F .F^xF)F1FBFFF]sF)**\"$n&F.F^xF)F1F;FFF2F)*(\")Ce;DF)F/F.FFF;F.**F\\yF. FaxF)F1FhnFFF7F)**\"#@F.F^xF)F1F2FFF;F)*,FgwF.FivF)F5F.F1F;FFF.F.*,\"$ 0%F.FdwF)FFFBF1F7F/F.F.*&\",'\\e?)z&F)FFFjtF)*&\"$G\"F)F/F2F.*(F9F.Fbw F)F1FjtF)**\"#kF)FFF.F5F.F/FBF.**F[xF)FFF7F5FBF1F.F)*,F2F.F[xF)FFF2F5F BF1FBF)*,F2F.F[xF)FFFBF5FBF1F2F)*.FjtF.F[xF)F5F.F1F2F/F.FFF.F)**F\\wF) F5F.FFF;F1F.F.*,F[wF.FdwF)FFF7F1FBF/F.F.**\"#'*F)F/F.FFFBF5FBF.*,FfxF. FivF)F5F.F1FBFFFhnF.*.F2F.F[xF)F5F.FFF2F/F.F1F.F)*,F`rF.F_wF)F1FBFFF.F /FBF)*(\")O([x$F)F5F.FFF]sF.F)#\"#8FBF.*,\"aowdjq**RV./Ut([l!)*yEL([3X C0/x'*[p&41l/.cF)FaqF;,8*&F_wF.FIF.F)**\"&gP&F.F5F.F1F.FFF.F)*(\"&c]%F .F5F.FLF.F.*(\"&SI#F.F5F.FOF.F.*(\"&o\"RF.F/F.FFF.F.*(\"&_*HF.F/F.F1F. F)*(\"%%G\"F.FEF.F1F.F.*(\"%-VF.FOF.FLF.F)*(\"%#\\$F.FFF.F0F.F.*&\"$t( F.FboF.F)*&\"$\")*F.F6F.F)F.F]qF;Ffv!\"'F.*,\"ensU@oQ^8iKLPBCyI\\\\*f8 .+DVJI6K,#F)FaqFhn,0*(\"$o(F.F5F.F1F.F)*&\"#bF.FEF.F)*&\"#vF.F0F.F.*( \"$0\"F.FLF.F1F.F.*(F`xF.FOF.FFF.F)*&Fh\\lF.F/F.F.*(\"%!G\"F.F5F.FFF.F .F.F]qFhnFfv!\"&F.*,\"O/VJX:#>^bRNUX2\"*4$>hK+;SVF)FaqF7,**&FhyF.F5F.F )*(\"#UF.F1F.FFF.F)*&\"#LF.FOF.F.*&\"#DF.FLF.F.F.F]qF7Ffv!\"%F)*,\"BW< B(yioUyC8%QhRsz(F)FaqF2,&FFF.F1F)F.F]qF2Ffv!\"$F)**\"7;?ylNV2\">>O$F)F aqFBF]qFBFfv!\"#F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "numerator=0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 755 "factor(243/4194304 *g^5*a*e+45/2097152*a^3*g^3*e+3/4194304*a^5*e*g-27/32768*e^2*g^3-2187/ 2147483648*g^8*a+243/8388608*g^6*e-1/2147483648*a^8*g+81/2097152*b*g^5 *a^2-1/73728*a^5*b^2-9/8192*b*g^2*a^2*e-9/32768*e^2*a^2*g-729/53687091 2*g^7*a^2-189/1073741824*g^4*a^5+1/432*b^3*a^3-1/32768*e^2*a^3+15/1048 576*b*g^3*a^4+135/4194304*b*g^4*a^3-1/24576*b*a^4*e-3/536870912*g^2*a^ 7-567/536870912*g^6*a^3+1/25165824*e*a^6-567/1073741824*g^5*a^4-21/536 870912*g^3*a^6+81/4194304*b*g^6*a+405/8388608*a^2*g^4*e-1/57982058496* a^9+1/128*e^3-729/2147483648*g^9+1/64*a*b*e^2-1/8192*a^4*b^2*g-3/8192* a^3*b^2*g^2-3/8192*a^2*b^2*g^3-9/8192*b*g^3*e*a+1/2097152*b*a^6*g+45/8 388608*a^4*g^2*e+1/96*e*a^2*b^2+15/4194304*b*g^2*a^5-3/8192*b*a^3*e*g- 27/32768*g^2*a*e^2+1/37748736*b*a^7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\",'\\e?)z&!\"\",.*&\"$o(\"\"\"%\"eGF*F**(\"$7&F*%\"bGF*%\"aGF* F**$)F/\"\"$F*F&*(\"\"*F*)F/\"\"#F*%\"gGF*F&*&\"#FF*)F7F2F*F&*(F9F*)F7 F6F*F/F*F&F2F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "solve(768 *e+512*b*a-a^3-9*a^2*g-27*g^3-27*g^2*a,e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"\"#\"\"$\"\"\"*&%\"bGF(%\"aGF(F(!\"\"*&#F(\"$o(F (*$)F+F'F(F(F(*&#F'\"$c#F(*&)F+F&F(%\"gGF(F(F(*&#\"\"*F4F(*$)F7F'F(F(F (*&F9F(*&)F7F&F(F+F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "D2:=eval(D1,e=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#D2G,<*&),&*$) %\"xG\"\"#\"\"\"F-%\"yGF-F,F-,&F)F-F.!\"\"F-F-*(%\"aGF-F+F-)F.\"\"$F-F -*&%\"bGF-)F.\"\"%F-F-*(,&F2F-%\"gGF-F-)F+F4F-)F.F,F-F-*(,,*&\"\")F0F; F,F0*(\"#;F0,.*&#\"#6\"\"'F-*&)F2F,F-F;F-F-F-*&#\"#:F,F-*&)F;F,F-F2F-F -F-*&#\"#F-F6F-F2F-FYF-F-**\"#kF-F6F- FTF-F;F-F0*(\"\"&F-)F2FgpF-F;F-F0*(\"#[F-F6F-)F2F8F-F-*(\"#!)F-F6F-)F; F8F-F0*(\"#9F-FTF-FYF-F-*(\"\"(F-FJF-F^qF-F0*(FioF-)F;FgpF-F2F-F0*$)F2 FHF-F-*&FHF-)F;FHF-F-*&\"%O:F-)F^oF,F-F-*&FjqF-),.*&#\"#DFcoF-FIF-F0*& #F-FfoF-FNF-F0*&#\"#BFcoF-FSF-F-*&#FfrFfoF-FXF-F-*&#F-F4F-FfnF-F0*&#F- F4F-FinF-F0F,F-F-*(\"$7\"F-F^oF-FTF-F-*(\"$O$F-F^oF-FYF-F-**\"$w\"F-F^ rF-FJF-F;F-F-*(\"%sIF-F^oF-F^rF-F0*(\"$S#F-F^rF-FYF-F0*(F]qF-F^rF-FTF- F0**\"%C5F-F6F-F^rF-F;F-F-**FisF-F6F-F^oF-F;F-F0**FisF-F^oF-F6F-F2F-F- **FisF-F2F-F^rF-F6F-F0**\"$W\"F-F^rF-FOF-F2F-F-**F^tF-F^oF-FJF-F;F-F0F -,(*(F,F-F;F-F2F-F0*$FOF-F-*$FJF-F-F0F.FgpF-*(F;F-)F+FgpF-F.F-F-*,FCF0 FDF-FjnF0F+F8F.F,F-*(F^rF-FF-FOF-F2F-F[rF-F0*&FiyF-)F^oF4F- F-*.FepF-F]pF-F`tF0F2F-F;F-F^oF-F-*(FbzF-F]rF-FYF-F-*(\"&_\"\\F-F[rF-F ^rF-F0*(F`_lF-F^oF-F]rF-F-**Fe[lF-F[qF-F[[lF-F;F-F-**\"%?^F-FTF-F[[lF- FOF-F0**Fd_lF-FJF-F[[lF-FYF-F-**F]qF-F6F-FfqF-F;F-F0*,FiyF-FOF-F2F-F^o F-F^rF-F-*,FbzF-FJF-F;F-F^rF-F^oF-F-*,F`vF-F6F-FTF-F^rF-F;F-F-*,F4F-F] pF-F`tF0F2F-F;F8F-*,F,F-F]pF-F`tF0F2F,F;F4F0*,\"&oF$F-F6F-F^oF-F^rF-F; F-F-*,F]`lF-F2F-F6F-F^oF-F^rF-F0*,FbwF-F6F-FYF-F^rF-F2F-F0*,FcuF-F]pF- F`tF0F;F,F^oF-F0*,FcuF-F]pF-F`tF0F;F,F^rF-F-*,FcuF-F]pF-F`tF0F2F,F^oF- F0*,FcuF-F]pF-F`tF0F2F,F^rF-F-*,FbzF-F6F-FTF-F^oF-F;F-F-**F_^lF-F6F-FO F-FhpF-F0*(FjpF-F6F-F]xF-F-F-,**(F4F-FJF-F;F-F0*(F4F-FOF-F2F-F-FSF-FXF 0F0F.FHF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomia l [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(D2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,$*(\")_:JG!\"\",2*&\"$#>\"\"\"%\"yGF* F&*&F)F*)%\"xG\"\"#F*F**(\"#kF*)F+F/F*%\"bGF*F**(\"\"*F*)%\"gGF/F*F2F* F***\"\"'F*F7F*%\"aGF*F2F*F***\"#'*F*F:F*F.F*F+F*F***FF*F/F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The polynomial fac tors, so case D is finished [5]. Now, case A." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "solve(32*e*a-160*e*g-2*a^3*g+4*g^2*a^2+2*a*g^3- a^4-3*g^4+32*j*a+96*j*g-64*b*g*a+32*b*g^2+32*b*a^2,e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#K!\"\",6*(\"\"#\"\"\"%\"aGF*)%\"gG\"\"$F*F& *$)F+\"\"%F*F**(F)F*)F+F.F*F-F*F**(F1F*)F-F)F*)F+F)F*F&*(\"#'*F*%\"jGF *F-F*F&**\"#kF*%\"bGF*F-F*F+F*F**&F.F*)F-F1F*F**(F%F*F9F*F+F*F&*(F%F*F " 0 "" {MPLTEXT 1 0 17 "A8:=eval(A7,e=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A8G,<*&),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-F,F-,&F)F-F.! \"\"F-F-*(%\"aGF-F+F-)F.\"\"$F-F-*&%\"bGF-)F.\"\"%F-F-*(,&F2F-%\"gGF-F -)F+F4F-)F.F,F-F-*(,,*&\"\")F0F;F,F0*(\"#;F0,2*&)F2F,F-F;F-F0*(\"\"&F- )F;F,F-F2F-F-*$)F2F4F-F0*&F4F-)F;F4F-F0*(F,F-,6*(F,F-F2F-FMF-F0*$)F2F8 F-F-*(F,F-FKF-F;F-F-*(F8F-FIF-FFF-F0*(\"#'*F-%\"jGF-F;F-F0**\"#kF-F6F- F;F-F2F-F-*&F4F-)F;F8F-F-*(\"#KF-FWF-F2F-F0*(FgnF-F6F-FIF-F0*(FgnF-F6F -FFF-F0F-,&F2F-*&FHF-F;F-F0F0F-*&FYF-FWF-F0*(FCF-F6F-F2F-F-*(FCF-F6F-F ;F-F0F-,&F2F-F;F0F0F-F6F-*(F8F0F;F-F2F-F-*&FAF0F2F,F0F-F*F-F3F-F-*,Fgn F0FOF-FjnF0F+F-F.F8F-**\"$7&F0,V**\"%C5F-F6F-FWF-F;F-F-**FVF-F6F-FFF-F IF-F0**\"$#>F-F6F-F2F-FMF-F-**\"$W\"F-FWF-FIF-F2F-F-**FYF-F6F-FKF-F;F- F0**\"$w\"F-FWF-FFF-F;F-F-*(\"#!)F-FWF-FKF-F0*(FHF-)F2FHF-F;F-F0*(\"#[ F-F6F-FRF-F-*(FapF-F6F-FenF-F0*(\"#9F-FKF-FMF-F-*(\"\"(F-FFF-FenF-F0*( \"\"*F-)F;FHF-F2F-F0*$)F2\"\"'F-F-*(\"$S#F-FWF-FMF-F0*&F`qF-)F;F`qF-F- *&\"%O:F-)FWF,F-F-*.F\\qF-F,F0FOF-FjnF0F2F,F;F-F0*.\"#>F-F,F0FOF-FjnF0 F;F,F2F-F0*,FgnF-FOF-FjnF0F6F-F2F-F-**FgoF-F2F-FWF-F6F-F0**F4F-F,F0FOF ,Fjn!\"#F-*,FjpF-F,F0FOF-FjnF0F2F4F-*,\"#@F-F,F0FOF-FjnF0F;F4F-**FVF-F OF-FjnF0FWF-F0*,FgnF-F6F-FOF-FjnF0F;F-F0F-,(*(F,F-F;F-F2F-F0*$FIF-F-*$ FFF-F-F0F.FHF-*(F;F-)F+FHF-F.F-F-*,FCF0FDF-F_oF0F+F8F.F,F-*(FWF-F " 0 "" {MPLTEXT 1 0 11 "factor(A8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**\"%[?!\"\",>*(\"#S\"\"\"%\"gGF*%\"yGF*F**(%\"a GF*)F,\"\"#F*)F+F0F*F&*()F.F0F*F/F*F+F*F&*&)F.\"\"$F*F/F*F**&F/F*)F+F6 F*F**(\"\")F*F.F*)%\"xGF0F*F**(\"#KF*F/F*%\"jGF*F&*(F)F*F;F*F+F*F&**\" #?F*F*(\"#!)F *F+F*F,F*F&F-F&F2F&F4F*F7F**(FKF*F.F*F;F*F**(F>F*F/F*F?F*F&*(FNF*F;F*F +F*F&**FBF*F