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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 }{PSTYLE "Map
le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 
1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "
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{SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT 257 29 "Chapter 16\n\nSolving \+
equations" }{TEXT 300 1 "\n" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 301 31 "\251 Copyright 2003 by Andr\351 Heck." }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT 270 2 "1." }{TEXT -1 165 " Compute the 6
th degree polynomial mapping of which the graphs goes through the poin
ts (-5,-120), (-3,-48), (-2,36), (1,120), (4,2400), (10,220380), and (
12,57408).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "with(CurveFitting);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#7*%(BSplineG%-BSplineCurveG%,Interacti
veG%-LeastSquaresG%8PolynomialInterpolationG%6RationalInterpolationG%'
SplineG%4ThieleInterpolationG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 82 "xdata := [-5,-3,-2,1,4,10,12]:\nydata := [-120, -48, 36, 129, 24
00, 220380, 57408]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Poly
nomialInterpolation(xdata, ydata, x);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,0*&#\"'TN=\"'![\\#\"\"\"*$)%\"xG\"\"'F(F(!\"\"*&#\"&hG'\"&uC\"F
(*$)F+\"\"&F(F(F(*&#\"'nH&*\"&Km\"F(*$)F+\"\"%F(F(F(*&#\"'6>hF1F(*$)F+
\"\"$F(F(F-*&#\")TicV\"&qB'F(*$)F+\"\"#F(F(F-*&#\"&'Rq\"$J#F(F+F(F-#\"
'gfx\"$$pF(" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Let us verify the result:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := unapply(%,x);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(,0*&#\"'TN=\"'![\\#\"\"\"*$)9$\"\"'F1F1!\"\"*&#\"&hG'\"&uC\"F1*$)F4
\"\"&F1F1F1*&#\"'nH&*\"&Km\"F1*$)F4\"\"%F1F1F1*&#\"'6>hF:F1*$)F4\"\"$F
1F1F6*&#\")TicV\"&qB'F1*$)F4\"\"#F1F1F6*&#\"&'Rq\"$J#F1F4F1F6#\"'gfx\"
$$pF1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(f, xdata
);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)!$?\"!#[\"#O\"$H\"\"%+C\"'!Q
?#\"&3u&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "% - ydata;\n" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"!F$F$F$F$F$F$" }}}{PARA 0 "" 0 
"" {TEXT -1 29 "Other forms of the polynomial" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "PolynomialInterpolation(xdata, ydata, x, form=La
grange);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0*2\"\"#\"\"\"\"%x8!\"
\",&%\"xGF&\"\"$F&F&,&F*F&F%F&F&,&F*F&F&F(F&,&F*F&\"\"%F(F&,&F*F&\"#5F
(F&,&F*F&\"#7F(F&F(*2F%F&\"$b%F(,&F*F&\"\"&F&F&F,F&F-F&F.F&F0F&F2F&F&*
0\"$_#F(F6F&F)F&F-F&F.F&F0F&F2F&F&*2\"#VF&\"%GrF(F6F&F)F&F,F&F.F&F0F&F
2F&F(*2\"#DF&\"$n&F(F6F&F)F&F,F&F-F&F0F&F2F&F&*2\"%tOF&\"%7UF(F6F&F)F&
F,F&F-F&F.F&F2F&F(*2\"$)fF&\"%XlF(F6F&F)F&F,F&F-F&F.F&F0F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "PolynomialInterpolation(xdat
a, ydata, x, form=Newton);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&
,&*&,&*&,&*&,&*(\"'TN=\"\"\"\"'![\\#!\"\"%\"xGF0F2#\"'&*\\9\"&Km\"F0F0
,&F3F0\"\"%F2F0F0#\"$\\%\"$o\"F0F0,&F3F0F0F2F0F0#\"#R\"\")F2F0,&F3F0\"
\"#F0F0F0\"#;F0F0,&F3F0\"\"$F0F0F0\"#OF0F0,&F3F0\"\"&F0F0F0\"$?\"F2" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 
"" {TEXT 271 2 "2." }{TEXT -1 74 " Check whether\n                    \+
                                       " }{XPPEDIT 18 0 "3*x^2+3*y^2+6
*x*y+6*x+6*y+2" "6#,.*&\"\"$\"\"\"*$%\"xG\"\"#F&F&*&F%F&*$%\"yGF)F&F&*
(\"\"'F&F(F&F,F&F&*&F.F&F(F&F&*&F.F&F,F&F&F)F&" }{TEXT -1 96 "\ncan be
 written in the form\n                                                \+
                    " }{XPPEDIT 18 0 "a*(x+b*y+c)^n+d" "6#,&*&%\"aG\"
\"\"),(%\"xGF&*&%\"bGF&%\"yGF&F&%\"cGF&%\"nGF&F&%\"dGF&" }{TEXT -1 24 
"\nfor suitable values of " }{TEXT 258 11 "a, b, c, d," }{TEXT -1 5 " \+
and " }{TEXT 259 1 "n" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 
"3*x^2 + 3*y^2 + 6*x*y + 6*x + 6*y + 2 =\n a*(x+b*y+c)^n + d ;\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.*&\"\"$\"\"\")%\"xG\"\"#F'F'*&F&F')
%\"yGF*F'F'*(\"\"'F'F)F'F-F'F'*&F/F'F)F'F'*&F/F'F-F'F'F*F',&*&%\"aGF')
,(F)F'*&%\"bGF'F-F'F'%\"cGF'%\"nGF'F'%\"dGF'" }}}{PARA 0 "" 0 "" 
{TEXT -1 61 "Because of equality of degrees in left- and right-hand si
de, " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 18 " must be equal to " }
{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "n := 2:" }}}{PARA 0 "" 0 "" {TEXT -1 70 "Consider th
e left-hand side minus the right-hand side as a polynomial." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "poly := collect((lhs - rhs)(%%), [x
,y], distributed);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%polyG,2*&,&
\"\"$\"\"\"%\"aG!\"\"F))%\"xG\"\"#F)F)*(,&\"\"'F)*(F.F)F*F)%\"bGF)F+F)
F-F)%\"yGF)F)*&,&F1F)*(F.F)F*F)%\"cGF)F+F)F-F)F)*&,&*&F*F))F3F.F)F+F(F
)F))F4F.F)F)*&,&F1F)**F.F)F*F)F8F)F3F)F+F)F4F)F)F.F)*&F*F))F8F.F)F+%\"
dGF+" }}}{PARA 0 "" 0 "" {TEXT -1 92 "The coefficients must be equal t
o 0. This leads to a system of equations that can be solved." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "coeffs(poly, \{x,y\});\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6(,(\"\"#\"\"\"*&%\"aGF%)%\"cGF$F%!\"\"%
\"dGF*,&\"\"'F%*(F$F%F'F%F)F%F*,&F-F%**F$F%F'F%F)F%%\"bGF%F*,&F-F%*(F$
F%F'F%F1F%F*,&\"\"$F%F'F*,&*&F'F%)F1F$F%F*F5F%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "solve(\{%\});\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<&/%\"aG\"\"$/%\"cG\"\"\"/%\"bGF)/%\"dG!\"\"" }}}{PARA 0 "" 0 "
" {TEXT -1 27 "Let us verify the solution." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "eval(poly, %);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT 272 2 "3." }{TEXT -1 69 " Consider the stoichiom
etry of the following two chemical reactions.\n" }{TEXT 273 3 "(a)" }
{TEXT -1 68 " For the values of p, q, r, s, t, u, and v is the reactio
n equation\n" }{XPPEDIT 18 0 "(p*K*Mn*O[4]+q*H[2]*S*O[4]+r*H[2]*C[2]*O
[4]) -> (s*K[2]*S*O[4]+t*Mn*S*O[4]+u*H[2]*O+v*C*O[2])" "6#f*6#,(**%\"p
G\"\"\"%\"KGF(%#MnGF(&%\"OG6#\"\"%F(F(**%\"qGF(&%\"HG6#\"\"#F(%\"SGF(&
F,6#F.F(F(**%\"rGF(&F26#F4F(&%\"CG6#F4F(&F,6#F.F(F(7\"6$%)operatorG%&a
rrowG6\",***%\"sGF(&F)6#F4F(F5F(&F,6#F.F(F(**%\"tGF(F*F(F5F(&F,6#F.F(F
(*(%\"uGF(&F26#F4F(F,F(F(*(%\"vGF(F=F(&F,6#F4F(F(FEFEFE" }{TEXT -1 20 
"         \nbalanced?\n" }{TEXT 274 3 "(b)" }{TEXT -1 120 " For what v
alues of p, q, r, s, and t is the reaction equation\n                 \+
                                        " }{XPPEDIT 18 0 "(p*C*O+q*C*O
[2]+r*H[2]) -> (s*C*H[4]+t*H[2]*O)" "6#f*6#,(*(%\"pG\"\"\"%\"CGF(%\"OG
F(F(*(%\"qGF(F)F(&F*6#\"\"#F(F(*&%\"rGF(&%\"HG6#F/F(F(7\"6$%)operatorG
%&arrowG6\",&*(%\"sGF(F)F(&F36#\"\"%F(F(*(%\"tGF(&F36#F/F(F*F(F(F9F9F9
" }{TEXT -1 11 "\nbalanced?\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 294 3 "
(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "p*(K+Mn+4*O) + q*(2*H+S+4*O) + r*(
2*H+2*C+4*O) = \ns*(2*K+S+4*O) + t*(Mn+S+4*O) + u*(2*H+O) + v*(C+2*O);
\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*&%\"pG\"\"\",(%\"KGF'%#MnGF'
*&\"\"%F'%\"OGF'F'F'F'*&%\"qGF',(*&\"\"#F'%\"HGF'F'%\"SGF'*&F,F'F-F'F'
F'F'*&%\"rGF',(*&F2F'F3F'F'*&F2F'%\"CGF'F'*&F,F'F-F'F'F'F',**&%\"sGF',
(*&F2F'F)F'F'F4F'*&F,F'F-F'F'F'F'*&%\"tGF',(F*F'F4F'*&F,F'F-F'F'F'F'*&
%\"uGF',&*&F2F'F3F'F'F-F'F'F'*&%\"vGF',&F;F'*&F2F'F-F'F'F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "collect((lhs-rhs)(%), \{K,Mn
,O,H,S,C\});\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&,0*&\"\"%\"\"\"%
\"pGF(F(*&F'F(%\"qGF(F(*&F'F(%\"rGF(F(*&F'F(%\"sGF(!\"\"*&F'F(%\"tGF(F
0%\"uGF0*&\"\"#F(%\"vGF(F0F(%\"OGF(F(*&,&F)F(*&F5F(F/F(F0F(%\"KGF(F(*&
,&F)F(F2F0F(%#MnGF(F(*&,(*&F5F(F+F(F(*&F5F(F-F(F(*&F5F(F3F(F0F(%\"HGF(
F(*&,(F+F(F/F0F2F0F(%\"SGF(F(*&,&F6F0*&F5F(F-F(F(F(%\"CGF(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "coeffs(%, \{K,Mn,O,H,S,C\});
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(,0*&\"\"%\"\"\"%\"pGF&F&*&F%F&%
\"qGF&F&*&F%F&%\"rGF&F&*&F%F&%\"sGF&!\"\"*&F%F&%\"tGF&F.%\"uGF.*&\"\"#
F&%\"vGF&F.,&F'F&*&F3F&F-F&F.,&F'F&F0F.,(*&F3F&F)F&F&*&F3F&F+F&F&*&F3F
&F1F&F.,(F)F&F-F.F0F.,&F4F.*&F3F&F+F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "isolve(\{%\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<
)/%\"rG,$*&\"\"&\"\"\"%$_Z1GF)F)/%\"vG,$*&\"#5F)F*F)F)/%\"tG,$*&\"\"#F
)F*F)F)/%\"sGF*/%\"pGF2/%\"qG,$*&\"\"$F)F*F)F)/%\"uG,$*&\"\")F)F*F)F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(_Z1=1, %);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<)/%\"rG\"\"&/%\"vG\"#5/%\"tG\"\"#/%\"
sG\"\"\"/%\"pGF,/%\"qG\"\"$/%\"uG\"\")" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 61 " So, we have found the following chemical reaction equati
on: " }{XPPEDIT 18 0 "(2*K*Mn*O[4]+3*H[2]*S*O[4]+5*H[2]*C[2]*O[4]) -> \+
(K[2]*S*O[4]+2*Mn*S*O[4]+8*H[2]*O+10*C*O[2])" "6#f*6#,(**\"\"#\"\"\"%
\"KGF(%#MnGF(&%\"OG6#\"\"%F(F(**\"\"$F(&%\"HG6#F'F(%\"SGF(&F,6#F.F(F(*
*\"\"&F(&F26#F'F(&%\"CG6#F'F(&F,6#F.F(F(7\"6$%)operatorG%&arrowG6\",**
(&F)6#F'F(F4F(&F,6#F.F(F(**F'F(F*F(F4F(&F,6#F.F(F(*(\"\")F(&F26#F'F(F,
F(F(*(\"#5F(F<F(&F,6#F'F(F(FDFDFD" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 293 3 
"(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p*(C+O) + q*(C+2*O) + r*(2*H) = \ns
*(C+4*H) + t*(2*H+O);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"pG
\"\"\",&%\"CGF'%\"OGF'F'F'*&%\"qGF',&F)F'*&\"\"#F'F*F'F'F'F'*(F/F'%\"r
GF'%\"HGF'F',&*&%\"sGF',&F)F'*&\"\"%F'F2F'F'F'F'*&%\"tGF',&*&F/F'F2F'F
'F*F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect((lhs-r
hs)(%), \{C,O,H\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,(*&\"\"#
\"\"\"%\"qGF(F(%\"pGF(%\"tG!\"\"F(%\"OGF(F(*&,(%\"sGF,F)F(F*F(F(%\"CGF
(F(*&,(*&F'F(%\"rGF(F(*&\"\"%F(F0F(F,*&F'F(F+F(F,F(%\"HGF(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "coeffs(%, \{C,O,H\});\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%,(*&\"\"#\"\"\"%\"qGF&F&%\"pGF&%\"tG!
\"\",(%\"sGF*F'F&F(F&,(*&F%F&%\"rGF&F&*&\"\"%F&F,F&F**&F%F&F)F&F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isolve(\{%\});\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<'/%\"tG,&%$_Z1G\"\"\"*&\"\"#F(%$_Z2GF(F(/%
\"sG,&F'F(F+F(/%\"rG,&*&\"\"$F(F'F(F(*&\"\"%F(F+F(F(/%\"pGF'/%\"qGF+" 
}}}{PARA 0 "" 0 "" {TEXT -1 80 "An infinite number of solutions for th
e chemical reaction equation. Just choose " }{XPPEDIT 18 0 "_Z1;" "6#%
$_Z1G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "_Z2;" "6#%$_Z2G" }{TEXT -1 
31 " positive, relative prime, and " }{XPPEDIT 18 0 "_Z1 <= Z2;" "6#1%
$_Z1G%#Z2G" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "subs(\{_Z1=1,_Z2=0\}, %);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/
%\"qG\"\"!/%\"tG\"\"\"/%\"sGF)/%\"rG\"\"$/%\"pGF)" }}}{PARA 0 "" 0 "" 
{TEXT -1 59 "corresponds with the following chemical reaction equation
: " }{XPPEDIT 18 0 "(C*O+3*H[2]) -> (C*H[4]+H[2]*O)" "6#f*6#,&*&%\"CG
\"\"\"%\"OGF(F(*&\"\"$F(&%\"HG6#\"\"#F(F(7\"6$%)operatorG%&arrowG6\",&
*&F'F(&F-6#\"\"%F(F(*&&F-6#F/F(F)F(F(F4F4F4" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "subs(\{_Z1=0,_Z2=1\}, %%);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<'/%\"sG\"\"\"/%\"tG\"\"#/%\"rG\"\"%/%\"pG\"\"!/%\"qGF&
" }}}{PARA 0 "" 0 "" {TEXT -1 59 "corresponds with the following chemi
cal reaction equation: " }{XPPEDIT 18 0 "(C*O[2]+4*H[2]) -> (C*H[4]+2*
H[2]*O)" "6#f*6#,&*&%\"CG\"\"\"&%\"OG6#\"\"#F(F(*&\"\"%F(&%\"HG6#F,F(F
(7\"6$%)operatorG%&arrowG6\",&*&F'F(&F06#F.F(F(*(F,F(&F06#F,F(F*F(F(F6
F6F6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(\{_Z1=1,_Z2=1\}
, %%%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/%\"pG\"\"\"/%\"qGF&/%
\"tG\"\"$/%\"sG\"\"#/%\"rG\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 59 "corre
sponds with the following chemical reaction equation: " }{XPPEDIT 18 
0 "(C*O+C*O[2]+7*H[2]) -> (2*C*H[4]+3*H[2]*O)" "6#f*6#,(*&%\"CG\"\"\"%
\"OGF(F(*&F'F(&F)6#\"\"#F(F(*&\"\"(F(&%\"HG6#F-F(F(7\"6$%)operatorG%&a
rrowG6\",&*(F-F(F'F(&F16#\"\"%F(F(*(\"\"$F(&F16#F-F(F)F(F(F7F7F7" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "Not all \+
chemical reaction equations are from chemistry point of view acceptibl
e, but from stoichiometric point of view all solutions are possible." 
}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 275 2 "4." }{TEXT -1 33 " Solve the \+
following equation in " }{TEXT 260 1 "x" }{TEXT -1 4 " by " }{TEXT 0 
5 "solve" }{TEXT -1 5 " and " }{TEXT 0 6 "fsolve" }{TEXT -1 52 ".\n   \+
                                               " }{XPPEDIT 18 0 "48x^5
+8x^4-6x^3+114x^2-37x+18=0" "6#/,.*&\"#[\"\"\"*$%\"xG\"\"&F'F'*&\"\")F
'*$F)\"\"%F'F'*&\"\"'F'*$F)\"\"$F'!\"\"*&\"$9\"F'*$F)\"\"#F'F'*&\"#PF'
F)F'F3\"#=F'\"\"!" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 
"eq := 48*x^5 + 8*x^4 - 6*x^3 + 114*x^2 - 37*x + 18 = 0;\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,.*&\"#[\"\"\")%\"xG\"\"&F)F)*&\"\")F
))F+\"\"%F)F)*&\"\"'F))F+\"\"$F)!\"\"*&\"$9\"F))F+\"\"#F)F)*&\"#PF)F+F
)F5\"#=F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eq,
 x);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6',&#\"\"\"\"\"'F%*&^#F$F%\"\"
&#F%\"\"#F%,&F$F%*&^##!\"\"F&F%F)F*F%,(*&\"#7F0,&\"%q>F%*&\"#!*F%\"$z%
F*F%#F%\"\"$F0*(F)F%F&F0F4#F0F:F0#F%F&F0,**&\"#CF0F4F9F%*(F)F%F3F0F4F<
F%#F%F&F0*(^#F*F%F:F*,&*&F3F0F4F9F0*(F)F%F&F0F4F<F%F%F%,**&F@F0F4F9F%*
(F)F%F3F0F4F<F%#F%F&F0*(^##F0F+F%F:F*FEF%F%" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 15 "fsolve(eq, x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$!+v!)eN:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Only real solutions. \+
Add the keyword " }{TEXT 0 7 "complex" }{TEXT -1 27 " for all numeric \+
solutions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(eq, x, \+
complex);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'$!+v!)eN:!\"*^$$\"+nmmm
;!#5$!+i*zns$F)^$F'$\"+i*zns$F)^$$\"+u.%z<&F)$!+[D8%4\"F%^$F0$\"+[D8%4
\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 
0 "" 0 "" {TEXT 276 2 "5." }{TEXT -1 37 " Solve the following equation
 in x.\n\n" }{TEXT 277 3 "(a)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x+1)^(
x+a)=(x+1)^2" "6#/),&%\"xG\"\"\"F'F',&F&F'%\"aGF'*$,&F&F'F'F'\"\"#" }
{TEXT -1 1 "\n" }{TEXT 278 3 "(b)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x+(
1+x)^(1/2)+(2+x)^(1/3)+(3+x)^(1/4)=5" "6#/,*%\"xG\"\"\"),&F&F&F%F&*&F&
F&\"\"#!\"\"F&),&F*F&F%F&*&F&F&\"\"$F+F&),&F/F&F%F&*&F&F&\"\"%F+F&\"\"
&" }{TEXT -1 1 "\n" }{TEXT 279 3 "(c)" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"2*arctan*x=arctan(2*x/(1-x^2))" "6#/*(\"\"#\"\"\"%'arctanGF&%\"xGF&-F
'6#*(F%F&F(F&,&F&F&*$F(F%!\"\"F." }{TEXT -1 1 "\n" }{TEXT 280 3 "(d)" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "2-sin(1-x)=2*x" "6#/,&\"\"#\"\"\"-%$si
nG6#,&F&F&%\"xG!\"\"F,*&F%F&F+F&" }{TEXT -1 1 "\n" }}{SECT 1 {PARA 0 "
" 0 "" {TEXT 266 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq := (x+1)^(x+a)
 = (x+1)^2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/),&%\"xG\"\"\"
F)F),&F(F)%\"aGF)*$)F'\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "solve(eq, x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!,&\"\"#
\"\"\"%\"aG!\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 31 "Three solutions. T
est validity:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "seq(testeq(
subs(x=s, eq)), s=%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%%trueGF#F#
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "
" 0 "" {TEXT 267 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq := x + (1+x)^(
1/2) + (2+x)^(1/3) + (3+x)^(1/4) = 5;\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#eqG/,*%\"xG\"\"\"*$,&F(F(F'F(#F(\"\"#F(*$),&F,F(F'F(#F(\"\"$F
(F(*$),&F1F(F'F(#F(\"\"%F(F(\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "solve(eq, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&\"
\"\"!\"\"*$)-%'RootOfG6$,T*&\"+#fG$zUF$%#_ZGF$F%*&\"*c2q#QF$)F.\"\"#F$
F%\"+H&Rf7#F$*&\"+55w.=F$)F.\"\"%F$F%*&\"+Ge\"Qv&F$)F.\"\"$F$F$*&\"+RV
\\.<F$)F.\"\"'F$F$*&\"+k+=\"e$F$)F.\"\"&F$F%*&\"*Q&RYyF$)F.\"\")F$F%*&
\"+wd)[Q\"F$)F.\"\"(F$F$*&\"*$Ho#G#F$)F.\"#5F$F$*&\"*wCvy$F$)F.\"\"*F$
F%*&\")%QUa%F$)F.\"#7F$F%*&\")Oc5zF$)F.\"#6F$F$*&\"(#eCjF$)F.\"#9F$F$*
&\")%)[98F$)F.\"#8F$F%*&\"'\"Q.'F$)F.\"#;F$F%*&\"(C)R<F$)F.\"#:F$F$*&
\"&$\\OF$)F.\"#=F$F$*&\"'7k<F$)F.\"#<F$F%*&\"%&3\"F$)F.\"#?F$F%*&\"&3F
\"F$)F.\"#>F$F$*&F?F$)F.\"#AF$F%*&\"$s&F$)F.\"#@F$F%*$)F.\"#CF$F$*&FWF
$)F.\"#BF$F$$\"+?%RON\"!\"*F2F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 21 "App
roximate solution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(
%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*z'RB$)!\"*" }}}{PARA 0 "
" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "eval(eq, x=%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
$\"+,+++]!\"*\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 0 "" 0 "" {TEXT 268 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 
"eq := 2*arctan(x) = arctan((2*x)/(1-x^2));\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#eqG/,$*&\"\"#\"\"\"-%'arctanG6#%\"xGF)F)-F+6#,$*(F(F
)F-F),&F)F)*$)F-F(F)!\"\"F5F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "solve(eq, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}
{PARA 0 "" 0 "" {TEXT -1 61 "This Maple answers indicates that the equ
ality holds for any " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 21 ". Let
 us verify this." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diff((lh
s-rhs)(eq), x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\",&
F&F&*$)%\"xGF%F&F&!\"\"F&*&,&*&F%F&,&F&F&F(F+F+F&*(\"\"%F&F*F%F/!\"#F&
F&,&F&F&*(F1F&F*F%F/F2F&F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "normal(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 
0 "" 0 "" {TEXT -1 125 "So the left- and right-hand side can only diff
er a constant. This constant is equal to 0 as the following substituti
on shows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(eq, x=0);
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 269 3 "(d)
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eq := 2-sin(1-x) = 2*x;\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,&\"\"#\"\"\"-%$sinG6#,&F(!\"\"%\"xGF
(F(,$*&F'F(F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(
eq, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" 
{TEXT -1 23 "A graphical validation:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "plot((lhs-rhs)(eq), x=-6..8);\n" }}{PARA 13 "" 1 "" 
{GLPLOT2D 368 368 368 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$!\"'\"\"!$\"3>@
\"G,M,VL\"!#;7$$!3+nmmT)R[p&!#<$\"3ML/O\\`&*)H\"F-7$$!3#HL$e>;KHaF1$\"
3ST[^^EIr7F-7$$!3Lmm;4'=28&F1$\"3)*Q^L*QJ8C\"F-7$$!3vmm;ki8I[F1$\"3d+u
#Q:)z47F-7$$!3+LLeMD)4`%F1$\"3)p.N=R@X<\"F-7$$!3)om;HtGOD%F1$\"3Z#)>o3
GVO6F-7$$!3%)***\\i$\\WmRF1$\"3Ok^x'3z+4\"F-7$$!3Mmm\"H/R%pOF1$\"3Q<[!
ze&zL5F-7$$!3+++D^cQtLF1$\"3)40)fwy&)*o*F17$$!3%HLL$[$e)oIF1$\"3&\\<jB
)*)pP*)F17$$!3Tmm;M0j+GF1$\"3t44ypvh8#)F17$$!3Q*****\\ap')\\#F1$\"3%G!
G6%4woM(F17$$!3')*****\\noa>#F1$\"3y*oQ:J(yWkF17$$!3q*****\\XyK!>F1$\"
3Og;\\.@\\qbF17$$!3]mm\"HuTzj\"F1$\"3y*ooR<dKz%F17$$!3(HLL$G2VA8F1$\"3
<VL+OwG9RF17$$!3oKLL8::b5F1$\"3M=*pSkF`A$F17$$!3&z***\\iCUUu!#=$\"3%\\
ftZc)[.DF17$$!3SMLLL'[.p%Faq$\"3c6$Ry&HCV>F17$$!3a#***\\iu)3n\"Faq$\"3
5`ZdDmc99F17$$\"3%4,+Dh`V?\"Faq$\"3))=a`D'ym))*Faq7$$\"3%Rnm;/mV?%Faq$
\"3#eSW<5\"o9hFaq7$$\"3ksm;aRJfpFaq$\"3?sagFdK(3$Faq7$$\"3/KLLeu*3$**F
aq$\"3f1*)\\i\"4.\"p!#?7$$\"3_LL3dPv,8F1$!3i()p*oMBJ1$Faq7$$\"3Q++D'oY
/d\"F1$!3sZ3D?J&)3gFaq7$$\"3#RLL3TU1'=F1$!3YAD!G#GBI'*Faq7$$\"3'******
**)HWg@F1$!3>rExdg!RS\"F17$$\"3w++]n$RPX#F1$!3&)=IN\"3AV\">F17$$\"37,+
v$p=vt#F1$!3%)z#HzR.*)[#F17$$\"3k****\\_sg_IF1$!3164(RGd!>KF17$$\"3olm
mO$GdL$F1$!3AwZ)=BF+&RF17$$\"3t,++D0-QOF1$!3'pmf!RP[$z%F17$$\"3qKL3x@%
>\"RF1$!3o*4IQ;Zif&F17$$\"3*>++]*3T6UF1$!3[h,sALe#\\'F17$$\"3ImmT?w=$
\\%F1$!33Ql\")z3xItF17$$\"3[++v)[Dxy%F1$!3!=Lp$yUbx\")F17$$\"3'pmm;4!p
v]F1$!3)3u'*e(HWb*)F17$$\"3M***\\PMirP&F1$!3!pgS<Yg')p*F17$$\"3oNLL`f^
ncF1$!3n2;8GDSL5F-7$$\"31ML$eXWW'fF1$!3g]aTv!H(*3\"F-7$$\"3momTSU\"*ei
F1$!33_N[0k@P6F-7$$\"3&=+++R,&HlF1$!3QJ#y*QKLu6F-7$$\"3lnm;*zC'RoF1$!3
8')yAt2%3@\"F-7$$\"3#RLLL(G+<rF1$!3Ih5qLC%*R7F-7$$\"3L++v`du7uF1$!3gU?
=A$H'p7F-7$$\"3=,+DE%4ep(F1$!3G(yWlUg!*H\"F-7$$\"\")F*$!3>@\"G,M,VL\"F
--%'COLOURG6&%$RGBGF*F*F*-%+AXESLABELSG6$Q\"x6\"Q!Fc[l-%%VIEWG6$;F(Fgz
%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT 281 2 "6." }{TEXT -1 28 " The Cartesian coordina
tes (" }{XPPEDIT 18 0 "(x,y,z)" "6%%\"xG%\"yG%\"zG" }{TEXT -1 52 ") ca
n easily be expressed in spherical coordinates (" }{XPPEDIT 18 0 "r,th
eta,phi" "6%%\"rG%&thetaG%$phiG" }{TEXT -1 66 ").\n                   \+
                                            " }{XPPEDIT 18 0 "x=r*cos*
theta*sin*phi" "6#/%\"xG*,%\"rG\"\"\"%$cosGF'%&thetaGF'%$sinGF'%$phiGF
'" }{TEXT -1 64 "\n                                                   \+
            " }{XPPEDIT 18 0 "y=r*sin*theta*sin*phi" "6#/%\"yG*,%\"rG
\"\"\"%$sinGF'%&thetaGF'F(F'%$phiGF'" }{TEXT -1 64 "\n                \+
                                               " }{XPPEDIT 18 0 "z=r*c
os*phi" "6#/%\"zG*(%\"rG\"\"\"%$cosGF'%$phiGF'" }{TEXT -1 58 "\nExpres
s the spherical coordinates in the Cartesian ones.\n" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "eqns := \{x=r*cos(theta)*sin(phi), \n         y=r*sin
(theta)*sin(phi),\n         z=r*cos(phi)\}; \n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqnsG<%/%\"xG*(%\"rG\"\"\"-%$cosG6#%&thetaGF*-%$sinG
6#%$phiGF*/%\"yG*(F)F*-F0F-F*F/F*/%\"zG*&F)F*-F,F1F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "solve(eqns, \{r,theta,phi\});\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<%/%\"rG-%'RootOfG6#,**$)%\"yG\"\"#\"\"\"!\"
\"*$)%\"xGF-F.F/*$)%\"zGF-F.F/*$)%#_ZGF-F.F./%$phiG-%'arctanG6$-F'6#,(
*&F7F.,(F*F.F0F.F3F.F.F.F*F/F0F/*&F5F.F&F//%&thetaG-F<6$*(F,F.F&F/F>F/
*(F2F.F&F/F>F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(%
,radical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%$phiG-%'arctanG6$*$,
$*&,&*$)%\"yG\"\"#\"\"\"!\"\"*$)%\"xGF0F1F2F1,(F-F1F3F1*$)%\"zGF0F1F1F
2F2#F1F0*&F9F1F6#F2F0/%\"rG*$F6F:/%&thetaG-F'6$*(F/F1F6F<F*F<*(F5F1F6F
<F*F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%,symboli
c);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%$phiG-%'arctanG6$*&,(*$)%
\"yG\"\"#\"\"\"F/*$)%\"xGF.F/F/*$)%\"zGF.F/F/#!\"\"F.,&F+F/F0F/#F/F.*&
F5F/F*F6/%&thetaG-F'6$*&F-F/F8F6*&F2F/F8F6/%\"rG*$F*F9" }}}{PARA 0 "" 
0 "" {TEXT -1 142 "\nNow with help of the user and considering it as a
 system of polynomial equations in the radius and the trigonometric fu
nctions of the angles." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "po
lys := map(lhs-rhs, eqns);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&polys
G<%,&%\"xG\"\"\"*(%\"rGF(-%$cosG6#%&thetaGF(-%$sinG6#%$phiGF(!\"\",&%
\"yGF(*(F*F(-F0F-F(F/F(F3,&%\"zGF(*&F*F(-F,F1F(F3" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 188 "polys := subs(\{cos(phi)=cf, sin(phi)=sf, c
os(theta)=ct, sin(theta)=st, tan(theta)=tt\}, [op(polys), cos(theta)^2
+sin(theta)^2-1, cos(phi)^2+sin(phi)^2-1, tan(theta)*cos(theta)-sin(th
eta)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&polysG7(,&%\"xG\"\"\"*(%
\"rGF(%#ctGF(%#sfGF(!\"\",&%\"yGF(*(F*F(%#stGF(F,F(F-,&%\"zGF(*&F*F(%#
cfGF(F-,(*$)F+\"\"#F(F(*$)F1F9F(F(F(F-,(*$)F5F9F(F(*$)F,F9F(F(F(F-,&*&
%#ttGF(F+F(F(F1F-" }}}{PARA 0 "" 0 "" {TEXT -1 37 "We compute a suitab
le Groebner basis:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(G
roebner):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "gbasis(polys, \+
plex(ct, st, cf, tt, sf, r));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7(,
**$)%\"rG\"\"#\"\"\"F)*$)%\"zGF(F)!\"\"*$)%\"yGF(F)F-*$)%\"xGF(F)F-,,*
&)%#sfGF(F)F/F)F)*&F6F)F2F)F)*&F6F)F+F)F)F.F-F1F-,&*&F3F)%#ttGF)F)F0F-
,**&%#cfGF)F/F)F)*&F?F)F2F)F)*&F?F)F+F)F)*&F,F)F'F)F-,(*&%#stGF)F/F)F)
*&FEF)F2F)F)*(F0F)F7F)F'F)F-,(*&%#ctGF)F/F)F)*&FJF)F2F)F)*(F3F)F7F)F'F
)F-" }}}{PARA 0 "" 0 "" {TEXT -1 33 "Select the first three equations:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "%[1..3];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%,**$)%\"rG\"\"#\"\"\"F)*$)%\"zGF(F)!\"\"*$)%\"yG
F(F)F-*$)%\"xGF(F)F-,,*&)%#sfGF(F)F/F)F)*&F6F)F2F)F)*&F6F)F+F)F)F.F-F1
F-,&*&F3F)%#ttGF)F)F0F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "
solve(\{op(%)\}, \{r,sf,tt\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%
/%#ttG*&%\"yG\"\"\"%\"xG!\"\"/%\"rG-%'RootOfG6$,**$)F'\"\"#F(F**$)F)F3
F(F**$)%\"zGF3F(F**$)%#_ZGF3F(F(/%&labelG%$_L1G/%#sfG-F.6#,(*&F:F(,(F1
F(F4F(F6F(F(F(F1F*F4F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c
onvert(%, radical);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%#ttG*&%\"
yG\"\"\"%\"xG!\"\"/%\"rG*$,(*$)F'\"\"#F(F(*$)F)F1F(F(*$)%\"zGF1F(F(#F(
F1/%#sfG*$,$*&,&F/F*F2F*F(F.F*F*F7" }}}{PARA 0 "" 0 "" {TEXT -1 34 "So
 we find the following solution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 91 "r = sqrt(x^2+y^2+z^2); \ntheta = arctan(y/x); \nphi=arcsin(sqr
t(((x^2+y^2)/(y^2+x^2+z^2))));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
\"rG*$,(*$)%\"yG\"\"#\"\"\"F+*$)%\"xGF*F+F+*$)%\"zGF*F+F+#F+F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&thetaG-%'arctanG6#*&%\"yG\"\"\"%\"x
G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$phiG-%'arcsinG6#*$*&,&*$)
%\"yG\"\"#\"\"\"F/*$)%\"xGF.F/F/F/,(F+F/F0F/*$)%\"zGF.F/F/!\"\"#F/F." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 
0 "" {TEXT 282 2 "7." }{TEXT -1 18 " Solve the system " }{XPPEDIT 18 
0 "\{x^2+y^2=5,x*y=y^2+2\}" "6#<$/,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)\"\"&
/*&F'F)F+F),&*$F+F(F)F(F)" }{TEXT -1 6 " with " }{TEXT 0 5 "solve" }
{TEXT -1 5 " and " }{TEXT 0 6 "fsolve" }{TEXT 295 2 ".\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 39 "eqns := \{x^2 + y^2 = 5, x*y = y^2 + 2\};" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<$/,&*$)%\"xG\"\"#\"\"\"F,*$)%\"yGF+
F,F,\"\"&/*&F*F,F/F,,&F-F,F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "solve(eqns, \{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%
\"yG-%'RootOfG6#,(*&\"\"#\"\"\")%#_ZG\"\"%F,F,*$)F.F+F,!\"\"F/F,/%\"xG
,&*$)F&\"\"$F,F2*&#F8F+F,F&F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "allvalues(%, 'dependent');\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6&<$/%\"xG,&*&\"\")!\"\",&\"\"\"F+*&\"#J#F+\"\"#^#F+F+F+#\"\"$F/F+*(F2
F+\"\"%F)F*F.F)/%\"yG,$*&F/F)F*F.F)<$/F6,$*&F/F)F*F.F+/F%,&*&F(F)F*F1F
)*(F2F+F4F)F*F.F+<$/F6,$*&F/F),&F+F+*&^#F)F+F-F.F+F.F)/F%,&*&F(F)FEF1F
+*(F2F+F4F)FEF.F)<$/F6,$*&F/F)FEF.F+/F%,&*&F(F)FEF1F)*(F2F+F4F)FEF.F+
" }}}{PARA 0 "" 0 "" {TEXT -1 57 "Only complex solutions. The numerica
l approximations are:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq
(evalf(s), s=%);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&<$/%\"yG^$$!+:()
)>7*!#5$!+lAfHwF)/%\"xG^$$!+()GC-A!\"*$\"*f!GgJF1<$/F%^$$\"+:())>7*F)$
\"+lAfHwF)/F-^$$\"+()GC-AF1$!*f!GgJF1<$/F%^$F'F9/F-^$F/F?<$/F%^$F7F*/F
-^$F=F2" }}}{PARA 0 "" 0 "" {TEXT 0 6 "fsolve" }{TEXT -1 50 " will not
 find the solutions and with the keyword " }{TEXT 0 7 "complex" }
{TEXT -1 19 " only one solution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "fsolve(eqns, \{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'f
solveG6$<$/,&*$)%\"xG\"\"#\"\"\"F-*$)%\"yGF,F-F-\"\"&/*&F+F-F0F-,&F.F-
F,F-<$F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fsolve(eqns, \+
\{x,y\}, complex);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG^$$\"+
:())>7*!#5$!+kAfHwF)/%\"xG^$$\"+()GC-A!\"*$\"+\"f!GgJF)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 
283 2 "8." }{TEXT -1 68 " Solve the following system of polynomial equ
ations in the unknowns " }{TEXT 262 1 "x" }{TEXT -1 2 ", " }{TEXT 263 
1 "y" }{TEXT -1 6 ", and " }{TEXT 264 1 "z" }{TEXT -1 15 " over R (her
e, " }{TEXT 261 1 "a" }{TEXT -1 71 " is a real constant).\n           \+
                                    \{ " }{XPPEDIT 18 0 "z^2-x^2-y^2+2
*a*x+2*a*z-a^2" "6#,.*$%\"zG\"\"#\"\"\"*$%\"xGF&!\"\"*$%\"yGF&F**(F&F'
%\"aGF'F)F'F'*(F&F'F.F'F%F'F'*$F.F&F*" }{TEXT -1 55 " = 0,\n          \+
                                       " }{XPPEDIT 18 0 "y*z-a*y-a*x+a
^2" "6#,**&%\"yG\"\"\"%\"zGF&F&*&%\"aGF&F%F&!\"\"*&F)F&%\"xGF&F**$F)\"
\"#F&" }{TEXT -1 77 "                        = 0,\n                   \+
                             " }{XPPEDIT 18 0 "-2*a+x+y" "6#,(*&\"\"#
\"\"\"%\"aGF&!\"\"%\"xGF&%\"yGF&" }{TEXT -1 42 "                      \+
              = 0 \}\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "eqns := \{z^2-x^2-
y^2+2*a*x+2*a*z-a^2=0,\n         y*z-a*y-a*x+a^2=0,\n         -2*a+x+y
=0\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/,.*$)%\"zG\"\"#\"
\"\"F,*$)%\"xGF+F,!\"\"*$)%\"yGF+F,F0*(F+F,%\"aGF,F/F,F,*(F+F,F5F,F*F,
F,*$)F5F+F,F0\"\"!/,**&F3F,F*F,F,*&F5F,F3F,F0*&F5F,F/F,F0F7F,F9/,(*&F+
F,F5F,F0F/F,F3F,F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve
(eqns, \{x,y,z\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$<%/%\"zG*&-%'Roo
tOfG6$,&*$)%#_ZG\"\"#\"\"\"F/F/F//%&labelG%$_L1GF/%\"aGF//%\"yG,$F&!\"
\"/%\"xG,&*&F.F/F3F/F/F&F/<%/F%*&-F(6$,(F+F/F.F7*&F.F/F-F/F//F1%$_L2GF
/F3F//F5,&*&#F/F.F/F>F/F/F3F//F9,&F3F/*&#F/F.F/F>F/F7" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "map(allvalues, [%], 'dependent');" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&<%/%\"xG,&*&\"\"#\"\"\"%\"aGF*F**&
F+F*^#F*F*F*/%\"zGF,/%\"yG*&^#!\"\"F*F+F*<%/F/F2/F1F,/F&,&*&F)F*F+F*F*
F2F*<%/F/*&,&F*F4*$\"\"$#F*F)F*F*F+F*/F1,&*(F)F4F>F*F+F*F*F+F*/F&,&F+F
**(F)F4F>F*F+F*F4<%/F/*&,&F*F4F?F4F*F+F*/F1,&*(F)F4FKF*F+F*F*F+F*/F&,&
F+F**(F)F4FKF*F+F*F4" }}}{PARA 0 "" 0 "" {TEXT -1 28 "Let us verify al
l solutions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "seq(map(test
eq,simplify(subs(s,eqns))), s=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&<
#%%trueGF#F#F#" }}}{PARA 0 "" 0 "" {TEXT -1 147 "The same solutions wo
uld have been found if we had split the problem via Groebner basis tec
hnique into two simpler systems of polynomial equations." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "polys := map(lhs-rhs, eqns);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&polysG<%,**&%\"yG\"\"\"%\"zGF)F)*&%
\"aGF)F(F)!\"\"*&F,F)%\"xGF)F-*$)F,\"\"#F)F),.*$)F*F2F)F)*$)F/F2F)F-*$
)F(F2F)F-*(F2F)F,F)F/F)F)*(F2F)F,F)F*F)F)F0F-,(*&F2F)F,F)F-F/F)F(F)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Groebner[gsolve](%, \{x,y,
z\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$7%7%,(*&\"\"&\"\"\")%\"aG\"
\"#F)F)*(\"\"%F)F+F)%\"xGF)!\"\"*$)F/F,F)F),(*&F,F)F+F)F0F/F)%\"yGF),(
F/F0%\"zGF)*&F,F)F+F)F)-%%plexG6%F7F5F/<$,&*&F,F)F+F)F0F/F),&*&\"\"(F)
F+F)F0*&F,F)F/F)F)7%7%,(*&\"\"$F)F*F)F)*(\"\"'F)F+F)F/F)F0*&F,F)F2F)F)
F3,(*&F,F)F/F)F)F7F)*&F,F)F+F)F0F9<#F=" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 284 2 "9." }{TEXT 
-1 5 " Let " }{TEXT 265 1 "f" }{TEXT -1 31 " be the homogeneous polyno
mial " }{XPPEDIT 18 0 "x[0]^3+x[1]^3+x[2]^3+x[3]^3" "6#,**$&%\"xG6#\"
\"!\"\"$\"\"\"*$&F&6#F*F)F**$&F&6#\"\"#F)F**$&F&6#F)F)F*" }{TEXT -1 
56 ". It defines the Fermat surface in the projective space " }
{XPPEDIT 18 0 "P^3" "6#*$%\"PG\"\"$" }{TEXT -1 56 " as\n              \+
                                     \{" }{XPPEDIT 18 0 "x[0]" "6#&%\"
xG6#\"\"!" }{TEXT -1 1 ":" }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }
{TEXT -1 1 ":" }{XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" }{TEXT -1 1 ":
" }{XPPEDIT 18 0 "x[3]" "6#&%\"xG6#\"\"$" }{TEXT -1 1 " " }{TEXT 292 
1 "\316" }{TEXT -1 1 " " }{XPPEDIT 18 0 "P^3" "6#*$%\"PG\"\"$" }{TEXT 
-1 3 " | " }{XPPEDIT 18 0 "f(x[0],x[1],x[2],x[3])=0" "6#/-%\"fG6&&%\"x
G6#\"\"!&F(6#\"\"\"&F(6#\"\"#&F(6#\"\"$F*" }{TEXT -1 2 "\}\n" }{TEXT 
285 3 "(a)" }{TEXT -1 111 " There are 27 lines on the surface; use Map
le to determine these lines. Hint: the line through the two points\n \+
" }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 1 ":" }{XPPEDIT 
18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 ":" }{XPPEDIT 18 0 "x[2]" "
6#&%\"xG6#\"\"#" }{TEXT -1 1 ":" }{XPPEDIT 18 0 "x[3]" "6#&%\"xG6#\"\"
$" }{TEXT -1 6 " and :" }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }
{TEXT -1 1 ":" }{XPPEDIT 18 0 "y[2]" "6#&%\"yG6#\"\"#" }{TEXT -1 1 ":
" }{XPPEDIT 18 0 "y[3]" "6#&%\"yG6#\"\"$" }{TEXT -1 103 " is described
 in so-called Pl\374cker coordinates by\n                             \+
                        " }{XPPEDIT 18 0 "p^(ij)" "6#)%\"pG%#ijG" }
{TEXT -1 4 " := " }{XPPEDIT 18 0 "x[i]*y[j]-x[j]*y[i]" "6#,&*&&%\"xG6#
%\"iG\"\"\"&%\"yG6#%\"jGF)F)*&&F&6#F-F)&F+6#F(F)!\"\"" }{TEXT -1 8 " ,
 with " }{XPPEDIT 18 0 "i,j=0,1,2,3" "6'%\"iG/%\"jG\"\"!\"\"\"\"\"#\"
\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "i<>j" "6#0%\"iG%\"jG" }{TEXT 
-1 58 ".\nIn this coordinate system, a line is on the surface iff\n" }
{XPPEDIT 18 0 "f( p^`01`*u[1]+p^`02`*u[2]+p^`03`*u[3],p^`10`*u[0]+p^`1
2`*u[2]+p^`13`*u[3],p^`20`*u[0]+p^`21`*u[1]+p^`23`*u[3],p^`30`*u[0]+p^
`31`*u[1]+p^`32`*u[2])=0" "6#/-%\"fG6&,(*&)%\"pG%#01G\"\"\"&%\"uG6#F,F
,F,*&)F*%#02GF,&F.6#\"\"#F,F,*&)F*%#03GF,&F.6#\"\"$F,F,,(*&)F*%#10GF,&
F.6#\"\"!F,F,*&)F*%#12GF,&F.6#F5F,F,*&)F*%#13GF,&F.6#F;F,F,,(*&)F*%#20
GF,&F.6#FBF,F,*&)F*%#21GF,&F.6#F,F,F,*&)F*%#23GF,&F.6#F;F,F,,(*&)F*%#3
0GF,&F.6#FBF,F,*&)F*%#31GF,&F.6#F,F,F,*&)F*%#32GF,&F.6#F5F,F,FB" }
{TEXT -1 9 "\nfor all " }{XPPEDIT 18 0 "u[0],u[1],u[2],u[3]" "6&&%\"uG
6#\"\"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 11 " in R, and " }
{XPPEDIT 18 0 "p^`01`*p^`23`+p^`02`*p^`31`+p^`03`*p^`12` = 0" "6#/,(*&
)%\"pG%#01G\"\"\")F'%#23GF)F)*&)F'%#02GF))F'%#31GF)F)*&)F'%#03GF))F'%#
12GF)F)\"\"!" }{TEXT -1 1 "\n" }{TEXT 286 3 "(b)" }{TEXT -1 172 " Dete
rmine the singular points on the Fermat surface. Recall that a point i
s singular when the function values and all partial derivatives in thi
s point are equal to zero.\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 298 3 "(
a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := (x0, x1, x2, x3) -> x0^3 + x1^
3 + x2^3 + x3^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6&%#x0G%#x
1G%#x2G%#x3G6\"6$%)operatorG%&arrowGF+,**$)9$\"\"$\"\"\"F4*$)9%F3F4F4*
$)9&F3F4F4*$)9'F3F4F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
210 "alias(p^`01`=p01, p^`02`=p02, p^`03`=p03,\n      p^`10`=p10, p^`1
2`=p12, p^`13`=p13,\n      p^`20`=p20, p^`21`=p21, p^`23`=p23,\n      \+
p^`30`=p30, p^`31`=p31, p^`32`=p32,\n      u[0]=u0, u[1]=u1, u[2]=u2, \+
u[3]=u3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "f( p01*u1 + p
02*u2 + p03*u3,\n   -p01*u0 + p12*u2 + p13*u3, \n   -p02*u0 - p12*u1 +
 p23*u3,\n   -p03*u0 - p13*u1 - p23*u2 );\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,**$),(*&)%\"pG%#01G\"\"\"&%\"uG6#F+F+F+*&)F)%#02GF+&F-
6#\"\"#F+F+*&)F)%#03GF+&F-6#\"\"$F+F+F:F+F+*$),(*&F(F+&F-6#\"\"!F+!\"
\"*&)F)%#12GF+F2F+F+*&)F)%#13GF+F8F+F+F:F+F+*$),(*&F0F+F?F+FB*&FDF+F,F
+FB*&)F)%#23GF+F8F+F+F:F+F+*$),(*&F6F+F?F+FB*&FGF+F,F+FB*&FOF+F2F+FBF:
F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "collect(%, \{u||(0.
.3)\}, distributed);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,J*&,(*$))%
\"pG%#01G\"\"$\"\"\"!\"\"*$))F)%#02GF+F,F-*$))F)%#03GF+F,F-F,)&%\"uG6#
\"\"!F+F,F,*(,&*(F+F,)F)%#12GF,)F0\"\"#F,F-*(F+F,)F)%#13GF,)F4FAF,F-F,
)F7FAF,&F86#F,F,F,*(,&*(F+F,F>F,)F(FAF,F,*(F+F,)F)%#23GF,FEF,F-F,FFF,&
F86#FAF,F,*(,&*(F+F,FCF,FLF,F,*(F+F,FNF,F@F,F,F,&F86#F+F,FFF,F,*(,&*(F
+F,)F>FAF,F0F,F-*(F+F,)FCFAF,F4F,F-F,F7F,)FGFAF,F,*0\"\"'F,FNF,FCF,F4F
,FPF,F7F,FGF,F-*0FjnF,FNF,F>F,F0F,FVF,F7F,FGF,F,*(,&*(F+F,FenF,F(F,F-*
(F+F,)FNFAF,F4F,F-F,F7F,)FPFAF,F,*0FjnF,FCF,FVF,F>F,F(F,F7F,FPF,F-*(,&
*(F+F,FgnF,F(F,F-*(F+F,F`oF,F0F,F-F,)FVFAF,F7F,F,*&,(*$)F>F+F,F-F&F,*$
)FCF+F,F-F,)FGF+F,F,*(,&*(F+F,FNF,FgnF,F-*(F+F,F0F,FLF,F,F,FhnF,FPF,F,
*(,&*(F+F,FNF,FenF,F,*(F+F,F4F,FLF,F,F,FVF,FhnF,F,*(,&*(F+F,F`oF,FCF,F
-*(F+F,F@F,F(F,F,F,FGF,FaoF,F,*0FjnF,F4F,FVF,F0F,F(F,FGF,FPF,F,*(,&*(F
+F,F`oF,F>F,F-*(F+F,FEF,F(F,F,F,FgoF,FGF,F,*&,(F.F,FjoF,*$)FNF+F,F-F,)
FPF+F,F,*(,&*(F+F,FCF,FenF,F,*(F+F,F4F,F@F,F,F,FVF,FaoF,F,*(,&*(F+F,Fg
nF,F>F,F,*(F+F,FEF,F0F,F,F,FgoF,FPF,F,*&,(F2F,F\\pF,FbqF,F,)FVF+F,F," 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "coeffs(%, [u||(0..3)]);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "66,&*(\"\"$\"\"\")%\"pG%#12GF&))F(%#02
G\"\"#F&!\"\"*(F%F&)F(%#13GF&))F(%#03GF-F&F.,&*(F%F&F'F&))F(%#01GF-F&F
&*(F%F&)F(%#23GF&F2F&F.,&*(F%F&)F'F-F&F+F&F.*(F%F&)F0F-F&F3F&F.,(*$)F8
F%F&F.*$)F+F%F&F.*$)F3F%F&F.,&*(F%F&F0F&F7F&F&*(F%F&F;F&F*F&F&,&*(F%F&
FAF&F8F&F.*(F%F&)F;F-F&F+F&F.,&*(F%F&F;F&F?F&F&*(F%F&F3F&F7F&F&,&*(F%F
&FOF&F'F&F.*(F%F&F2F&F8F&F&,&*(F%F&F0F&F?F&F&*(F%F&F3F&F*F&F&,&*(F%F&F
AF&F'F&F&*(F%F&F2F&F+F&F&,&*(F%F&F?F&F8F&F.*(F%F&FOF&F3F&F.,&*(F%F&F;F
&FAF&F.*(F%F&F+F&F7F&F&,&*(F%F&FOF&F0F&F.*(F%F&F*F&F8F&F&,(*$)F'F%F&F.
FCF&*$)F0F%F&F.,(FEF&F`oF&*$)F;F%F&F.,(FGF&FboF&FeoF&,$**\"\"'F&F;F&F0
F&F3F&F.,$**FjoF&F;F&F'F&F+F&F&,$**FjoF&F0F&F'F&F8F&F.,$**FjoF&F3F&F+F
&F8F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "polys := \{%, p0
1*p23-p02*p13+p03*p12\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&polysG<
7,&*(\"\"$\"\"\")%\"pG%#12GF)))F+%#02G\"\"#F)!\"\"*(F(F))F+%#13GF)))F+
%#03GF0F)F1,&*(F(F)F*F)))F+%#01GF0F)F)*(F(F))F+%#23GF)F5F)F1,&*(F(F)F3
F)F:F)F)*(F(F)F>F)F-F)F),&*(F(F))F*F0F)F.F)F1*(F(F))F3F0F)F6F)F1,(*$)F
;F(F)F1*$)F.F(F)F1*$)F6F(F)F1,(FKF)*$)F*F(F)F)*$)F>F(F)F1,(FPF1FIF)*$)
F3F(F)F1,&*(F(F)FEF)F;F)F1*(F(F))F>F0F)F6F)F1,&*(F(F)FGF)F;F)F1*(F(F)F
ZF)F.F)F1,&*(F(F)F>F)FGF)F1*(F(F)F.F)F:F)F),&*(F(F)F>F)FEF)F)*(F(F)F6F
)F:F)F),&*(F(F)FZF)F3F)F1*(F(F)F-F)F;F)F),&*(F(F)FZF)F*F)F1*(F(F)F5F)F
;F)F),$**\"\"'F)F>F)F*F)F.F)F),$**FfoF)F>F)F3F)F6F)F1,&*(F(F)F3F)FEF)F
)*(F(F)F6F)F-F)F),&*(F(F)FGF)F*F)F)*(F(F)F5F)F.F)F),(FMF)FUF)FRF),(*&F
;F)F>F)F)*&F.F)F3F)F1*&F6F)F*F)F),$**FfoF)F3F)F*F)F;F)F1,$**FfoF)F6F)F
.F)F;F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sols := \{solv
e(polys)\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG<1<(/)%\"pG%#23
GF(/)F)%#12G\"\"!/)F)%#03GF./)F)%#01G,&*&-%'RootOfG6$,(*$)%#_ZG\"\"#\"
\"\"F?F?F?F=F?/%&labelG%$_L3GF?F(F?F?F(F?/)F)%#13G,$F6!\"\"/)F)%#02GF6
<(F'/F3,$F(FGF+F//FDF5FH<(F'/FDF./FIF./F,F(/F0FM/F3F(<(F'FPFQFS/F,F6/F
3F6<(F'FPFQFR/F0*&-F86$,(F;F?F?F?F=FG/FA%$_L2GF?F(F?/F3,$FZFG<(F'FPFQ/
F,F[o/F3,&FZF?F(FGFY<(F'FPFQ/F,F_oFTFY<(/F(F./FDFD/F3F./F,*&-F86$Fgn/F
A%$_L1GF?FDF?/FIFD/F0,&FgoFGFDF?<(F'FLF+F//FIF(/FDFM<(F'F+/FDFZ/F3FZF/
F`p<(F'F+F/FH/F3FFFap<(FcoFdoFeo/F0,$FDFGFfo/FI*&FDF?,&FhoF?F?FGFG<(Fc
oFdo/F0Fgo/FI,&FgoF?FDFGFeoFfo<(FcoFdoF^qFeo/F,Fip/FI,$FgoFG<(FcoFdoFe
oFhpFbqF\\p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "map(allvalue
s, %, dependent);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<=<(/)%\"pG%#23GF
&/)F'%#12G\"\"!/)F'%#03GF,/)F'%#02G*&,&#\"\"\"\"\"#!\"\"*&^##F6F7F6\"
\"$F;F6F6F&F6/)F'%#01G,$F3F8/)F'%#13G,$F&F8<(F%F)F-/F1F&/FB*&,&F;F6F9F
6F6F&F6/F>FH<(F%/FBF,/F1F,/F*F&/F.FD/F>F&<(F%/F>FDF)F-FFFA<(/F&F,/FBFB
/F>F,/F.,$FBF8/F*FX/F1FB<(F%F)F-/F>,&F3F6F&F6F0/FBF@<(/F>,&*&,&#F6F7F8
*&^##F8F7F6F<F;F6F6F&F6F6F&F6F%F)F-/FB,$F\\oF8/F1F\\o<(F%FLFMFO/F*F3/F
>F3<(F%FLFMFO/F*F\\o/F>F\\o<(F%FRF)F-F0/FBFgn<(F%FRF)F-Fdo/FBF[o<(F%FL
FMFN/F.FH/F>,$FHF8<(F%FLFMFN/F.*&,&F;F6F_oF6F6F&F6/F>,$FepF8<(F%FLFMF`
p/F*Fbp/F>,&FHF6F&F8<(F%FLFMFdp/F*Fhp/F>,&FepF6F&F8<(FTFUFV/F**&FIF6FB
F6/F.,&FcqF8FBF6FZ<(FTFUFV/F**&FfpF6FBF6/F.,&FhqF8FBF6FZ<(F%FLFMFPF`p/
F*F\\q<(F%FLFMFPFdp/F*F`q<(F%F)F-FF/FBFep/F>Fep<(F%F)F-Fdo/F>FcoFA<(FT
FUFVFWFbq/F1*&FBF6F4F8<(FTFUFVFWFgq/F1*&FBF6F]oF8<(FTFUFVFbq/F.Fcq/F1,
&FcqF6FBF8<(FTFUFVFgq/F.Fhq/F1,&FhqF6FBF8<(FTFUFVFYF[s/F1,$FcqF8<(FTFU
FV/F1,$FhqF8FYF_s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(%)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 299 3 "(b)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 51 "f := (x0, x1, x2, x3) -> x0^3 + x1^3 + x2^3 \+
+ x3^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6&%#x0G%#x1G%#x2G%#
x3G6\"6$%)operatorG%&arrowGF+,**$)9$\"\"$\"\"\"F4*$)9%F3F4F4*$)9&F3F4F
4*$)9'F3F4F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "eqns
 := \{f(w,x,y,z)=0, D[1](f)(w,x,y,z)=0, D[2](f)(w,x,y,z)=0, D[3](f)(w,
x,y,z)=0, D[4](f)(w,x,y,z)=0\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
eqnsG<'/,**$)%\"wG\"\"$\"\"\"F,*$)%\"xGF+F,F,*$)%\"yGF+F,F,*$)%\"zGF+F
,F,\"\"!/,$*&F+F,)F*\"\"#F,F,F6/,$*&F+F,)F/F;F,F,F6/,$*&F+F,)F5F;F,F,F
6/,$*&F+F,)F2F;F,F,F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sol
ve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"wG\"\"!/%\"yGF&/%\"zGF
&/%\"xGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT 287 3 "10." }{TEXT -1 40 " Solve the following i
nequalities in x.\n" }{TEXT 288 3 "(a)" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "abs(x-3)*abs(3-x)>abs(x)" "6#2-%$absG6#%\"xG*&-F%6#,&F'\"\"\"\"\"$!
\"\"F,-F%6#,&F-F,F'F.F," }{TEXT -1 1 "\n" }{TEXT 289 3 "(b)" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "abs(x^3-x^2-x-1)>1/abs(x^2-1)" "6#2*&\"\"\"F%
-%$absG6#,&*$%\"xG\"\"#F%F%!\"\"F--F'6#,**$F+\"\"$F%*$F+F,F-F+F-F%F-" 
}{TEXT -1 1 "\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 296 3 "(a)" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "abs(x-3)*abs(3-x) > abs(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#2-%$absG6#%\"xG*$)-F%6#,&F'\"\"\"\"\"$!\"\"\"\"#F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(%, \{x\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$<#2,&#\"\"(\"\"#\"\"\"*&F(!\"\"\"#8#F)F(F)%
\"xG<#2F.,&F&F)*&F(F+F,F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 297 3 "(b)" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "abs(x^3-x^2-x-1) > 1/abs(x^2-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#2*&\"\"\"F%-%$absG6#,&*$)%\"xG\"\"#F%F%F%!\"\"F.-F'6#,*
*$)F,\"\"$F%F%F*F.F,F.F%F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "solve(%, \{x\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&<#2-%'RootOfG6
$,**$)%#_ZG\"\"%\"\"\"F-*&\"\"#F-)F+F/F-!\"\"*$)F+\"\"$F-F1F-F-$\"+oh;
0>!\"*%\"xG<#2F8-F&6$,,F/F-*$)F+\"\"&F-F-*&F/F-F3F-F1F)F1F+F-$!+3G5m6F
7<$2-F&6$F=$\"+(H`&Q7F7F82F8-F&6$F=$\"+q[pT<F7<$2\"\"!F82F8-F&6$F($\"+
ngV5n!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(evalf,[%]);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&<#2$\"+oh;0>!\"*%\"xG<#2F)$!+3G5
m6F(<$2$\"+(H`&Q7F(F)2F)$\"+q[pT<F(<$2$\"\"!F8F)2F)$\"+ngV5n!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "
" {TEXT 290 3 "11." }{TEXT -1 31 " Solve the recurrence equation " }
{XPPEDIT 18 0 "a[n+1]=(8/5)*a[n]-a[n-1],a[0]=0,a[1]=1" "6%/&%\"aG6#,&%
\"nG\"\"\"F)F),&*(\"\")F)\"\"&!\"\"&F%6#F(F)F)&F%6#,&F(F)F)F.F./&F%6#
\"\"!F7/&F%6#F)F)" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 
"\{a(n+1) = 8/5*a(n) - a(n-1), a(0)=0, a(1)=1\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<%/-%\"aG6#,&%\"nG\"\"\"F*F*,&*&#\"\")\"\"&F*-F&6#F)F*F
*-F&6#,&F)F*F*!\"\"F5/-F&6#\"\"!F9/-F&6#F*F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "rsolve(%, a(n));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&^##!\"&\"\"'\"\"\")^$#\"\"%\"\"&#\"\"$F.%\"nGF)F)*&^##F.F(F)
)^$F,#!\"$F.F1F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"&\"\"$\"\"\"-%$sinG6#*
&%\"nGF(-%'arctanG6##F'\"\"%F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Le
t us verify the answer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A
 := unapply(%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGf*6#%\"nG6
\"6$%)operatorG%&arrowGF(,$*&#\"\"&\"\"$\"\"\"-%$sinG6#*&9$F1-%'arctan
G6##F0\"\"%F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A
(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "A(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A(n+1)-8/5*A(n)+A(n-1);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"&\"\"$\"\"\"-%$sinG6#*&,&%
\"nGF(F(F(F(-%'arctanG6##F'\"\"%F(F(F(*&#\"\")F'F(-F*6#*&F.F(F/F(F(!\"
\"*&F%F(-F*6#*&,&F.F(F(F:F(F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 
0 "" {TEXT 291 3 "12." }{TEXT -1 31 " Solve the recurrence equation " 
}{XPPEDIT 18 0 "a[n+1]=3*n*a[n]-2n(n-1)*a[n-1],a[1]=5,a[2]=54" "6%/&%
\"aG6#,&%\"nG\"\"\"F)F),&*(\"\"$F)F(F)&F%6#F(F)F)*(\"\"#F)-F(6#,&F(F)F
)!\"\"F)&F%6#,&F(F)F)F4F)F4/&F%6#F)\"\"&/&F%6#F0\"#a" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "\{a(n+1) =
 3*n*a(n) - 2*n*(n-1)*a(n-1), a(1)=5, a(2)=54\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<%/-%\"aG6#,&%\"nG\"\"\"F*F*,&*(\"\"$F*F)F*-F&6#F)F*F**
*\"\"#F*F)F*,&F)F*F*!\"\"F*-F&6#F2F*F3/-F&6#F*\"\"&/-F&6#F1\"#a" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(%, a(n));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*(\"#\\\"\"\")\"\"#,&%\"nGF&F&!\"\"F&-%&GA
MMAG6#F*F&F&*&\"#WF&F,F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%&GAMMAG6#%\"nG\"
\"\",&*&\"#\\F()\"\"#,&F'F(F(!\"\"F(F(\"#WF/F(" }}}{PARA 0 "" 0 "" 
{TEXT -1 25 "Let us verify the answer." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "A := unapply(%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"AGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&-%&GAMMAG6#9$\"\"\",&*&\"#
\\F1)\"\"#,&F0F1F1!\"\"F1F1\"#WF8F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "A(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A(2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A(n+
1) - 3*n*A(n) + 2*n*(n-1)*A(n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
(*&-%&GAMMAG6#,&%\"nG\"\"\"F*F*F*,&*&\"#\\F*)\"\"#F)F*F*\"#W!\"\"F*F**
*\"\"$F*F)F*-F&6#F)F*,&*&F-F*)F/,&F)F*F*F1F*F*F0F1F*F1*,F/F*F)F*F9F*-F
&6#F9F*,&*&F-F*)F/,&F)F*F/F1F*F*F0F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0" 0 }
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