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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 47 "Fractions rationnelles
\nAccompagnement - corrig\351" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 
"Exercice 1" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 3 "1.1" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "convert(1/(X^4+1),parfrac,X);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$%\"XG\"\"%\"\"\"F(F(!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(1/(X^4+1),parfrac,X,
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'F)%\"XGF'F'F',(*$F+F)F'F(!\"\"F'F'F.#F.\"\"%*&,&F)F'F(F'F',(F-F'F(F'F
'F'F.#F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "convert(1/(X^
4+1),parfrac,X,\{I,sqrt(2)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(
,&#\"\"\"\"\"%F'%\"IG#!\"\"F(F'\"\"##F'F,,(%\"XGF,*$F,F-F'*&F)F'F,F-F+
F+F'*(,&F*F'F)F&F'F,F-,(F/F,F0F+F1F'F+F'*(,&F*F'F)F*F'F,F-,(F/F,F0F+F1
F+F+F'*(,&F&F'F)F&F'F,F-,(F/F,F0F'F1F'F+F'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 6 "?alias" }}{PARA 15 "" 0 "" {TEXT -1 440 "Mathematics
 is full of special notations and abbreviations. These notations are t
ypically encountered in written material as statements like ``let J de
note the Bessel function of the first kind'' or ``let alpha denote a r
oot of the polynomial x^3-2''. The purpose of the alias facility is to
 allow the user to state such abbreviations for the longer unique name
s that Maple uses and, more generally, to give names to arbitrary expr
essions. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alias(alpha=Ro
otOf(X^4+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"IG%&alphaG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(1/(X^4+1),parfrac,X,
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G\"\"\"*$F%F&F)!\"\"#F)\"\"%*&F%F&,&F(F)F*F+F+#F+F-*&F%F),&F(F)F%F)F+F
,*&F%F),&F(F)F%F+F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "co
nvert(1/(X^4-1),parfrac,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$,&%
\"XG\"\"\"!\"\"F'F(#F'\"\"%*$,&F&F'F'F'F(#F(F**$,&*$F&\"\"#F'F'F'F(#F(
F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "alias(beta=RootOf(X^4
+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%\"IG%&alphaG%%betaG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "convert(1/(X^4-1),parfrac,X,
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\"\"\"*$F%F&F)!\"\"#F+\"\"%*&F%F&,&F(F)F*F+F+#F)F-*$,&F(F)F+F)F+F0*$,&
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "convert(1/(1+X+X^2),parfrac,X);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(*$%\"XG\"\"#\"\"\"F&F(F(F(!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "alias(j=RootOf(X^2+X+1));
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "convert(1/(1+X+X^2),parfrac,
X,\{j\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%\"jG\"\"#\"\"\"F(F
(,&%\"XGF(F&!\"\"F+#F+\"\"$*&F%F(,(F*F(F(F(F&F(F+#F(F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(1/(X^8-1),parfrac,X,sqrt(2)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&!\"#\"\"\"*&\"\"##F'F)%\"X
GF'F'F',(*$F+F)F'F(!\"\"F'F'F.#F'\"\")*&,&F)F'F(F'F',(F-F'F(F'F'F'F.#F
.F0*$,&F-F'F'F'F.#F.\"\"%*$,&F+F'F'F'F.F4*$,&F+F'F.F'F.F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alias(omega=RootOf(X^8-1));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6(%\"IG%&alphaG%%betaG%\"jG%&omagaG%&ome
gaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(1/(X^8-1),pa
rfrac,X,\{omega\});" }}{PARA 8 "" 1 "" {TEXT -1 172 "Error, (in evala)
 reducible RootOf detected.  Substitutions are, \{RootOf(_Z^8-1) = -1,
 RootOf(_Z^8-1) = RootOf(_Z^2+1), RootOf(_Z^8-1) = RootOf(_Z^4+1), Roo
tOf(_Z^8-1) = 1\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor
(X^8-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"XG\"\"\"!\"\"F&F&,&
F%F&F&F&F&,&*$F%\"\"#F&F&F&F&,&*$F%\"\"%F&F&F&F&" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "convert(1/(X^8-1),parfrac,X,\{alpha\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*&%&alphaG\"\"$,&%\"XG\"\"\"*$F%F&!
\"\"F+#F)\"\")*&F%F&,&F(F)F*F)F+#F+F-*&F%F),&F(F)F%F)F+F0*&F%F),&F(F)F
%F+F+F,*&F%\"\"#,&F(F)*$F%F6F)F+F0*&F%F6,&F(F)F8F+F+F,*$,&F(F)F+F)F+F,
*$,&F(F)F)F)F+F0" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 3 "1.3" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "convert(1/(X^2-2*cos(theta)+
1),parfrac,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(*$%\"XG\"\"#\"\"
\"-%$cosG6#%&thetaG!\"#F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "alias(alpha=RootOf(X^2-2*cos(theta)+1));" }}{PARA 8 "
" 1 "" {TEXT -1 48 "Error, (in RootOf) expression independent of, _Z" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "alias(alpha=RootOf(X^2-2*
cos(theta)+1,X));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%\"IG%&alphaG%%be
taG%\"jG%&omagaG%&omegaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 
"convert(1/(X^2-2*cos(theta)+1),parfrac,X,\{alpha\});" }}{PARA 8 "" 1 
"" {TEXT -1 97 "Error, (in factor) 2nd argument is not a valid algebra
ic extension, \{RootOf(_Z^2-2*cos(theta)+1)\}" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 9 "Mouais..." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Ex
ercice 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "convert(1/((X^2-
1)*(X^2+1)^2),parfrac,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$,&%\"
XG\"\"\"!\"\"F'F(#F'\"\")*$,&F&F'F'F'F(#F(F**$,&*$F&\"\"#F'F'F'F(#F(\"
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(1/((X+1)^3*(X^2+X+1)^2),parfrac,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,,*$,&%\"XG\"\"\"F'F'!\"$F'*$F%!\"#\"\"#*$F%!\"\"F'*&,&F&F'F+F'F',(*
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r " }{TEXT 257 1 "C" }{TEXT -1 16 " ca donnerait..." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 25 "alias(j=RootOf(X^2+X+1));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6(%\"IG%&alphaG%%betaG%\"jG%&omagaG%&omegaG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "convert(1/((X+1)^3*(X^2+X+1)
^2),parfrac,X,\{j\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&,&%\"jG\"
#8\"#6\"\"\"F),(%\"XGF)F)F)F&F)!\"\"#F,\"\"**$F*!\"##F)\"\"$*&,&F&F'\"
\"#F)F),&F+F)F&F,F,#F)F.*$F6F0F1*$,&F+F)F)F)F,F)*$F:F0F5*$F:!\"$F)" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F:=n->n!/product(X+k,k=1..
n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6#%\"nG6\"6$%)operatorG%
&arrowGF(*&-%*factorialG6#9$\"\"\"-%(productG6$,&%\"XGF1%\"kGF1/F7;F1F
0!\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for n from 1 \+
to 8 do print(convert(F(n),parfrac,X)) od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$,&%\"XG\"\"\"F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*$,&%\"XG\"\"\"F'F'!\"\"\"\"#*$,&F&F'F)F'F(!\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*$,&%\"XG\"\"\"F'F'!\"\"\"\"$*$,&F&F'\"\"#F'F(
!\"'*$,&F&F'F)F'F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$,&%\"XG\"
\"\"F'F'!\"\"\"\"%*$,&F&F'\"\"#F'F(!#7*$,&F&F'\"\"$F'F(\"#7*$,&F&F'F)F
'F(!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$,&%\"XG\"\"\"F'F'!\"\"
\"\"&*$,&F&F'\"\"#F'F(!#?*$,&F&F'\"\"$F'F(\"#I*$,&F&F'\"\"%F'F(F-*$,&F
&F'F)F'F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$,&%\"XG\"\"\"F'F'!
\"\"\"\"'*$,&F&F'\"\"#F'F(!#I*$,&F&F'\"\"$F'F(\"#g*$,&F&F'\"\"%F'F(!#g
*$,&F&F'\"\"&F'F(\"#I*$,&F&F'F)F'F(!\"'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,0*$,&%\"XG\"\"\"F'F'!\"\"\"\"(*$,&F&F'\"\"#F'F(!#U*$,&F&F'\"\"$
F'F(\"$0\"*$,&F&F'\"\"%F'F(!$S\"*$,&F&F'\"\"&F'F(F1*$,&F&F'\"\"'F'F(F-
*$,&F&F'F)F'F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*$,&%\"XG\"\"\"F
'F'!\"\"\"\")*$,&F&F'\"\"#F'F(!#c*$,&F&F'\"\"$F'F(\"$o\"*$,&F&F'\"\"%F
'F(!$!G*$,&F&F'\"\"&F'F(\"$!G*$,&F&F'\"\"'F'F(!$o\"*$,&F&F'\"\"(F'F(\"
#c*$,&F&F'F)F'F(!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq
(binomial(8,k),k=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\")\"#G\"
#c\"#qF%F$F#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq(k
*binomial(8,k),k=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\")\"#c\"
$o\"\"$!GF&F%F$F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "conver
t((X^5+2)/(X^2+X+1)^3,parfrac,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
(*&,&\"\"\"F&%\"XG!\"\"F&,(*$F'\"\"#F&F'F&F&F&!\"$F&*&,&\"\"$F&F'F&F&F
)!\"#F&*&,&F'F&F0F&F&F)F(F&" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Su
r C..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "convert((X^5+2)/(X
^2+X+1)^3,parfrac,X,\{j\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&,&%
\"jG\"\"\"\"\"&F'F',(%\"XGF'F'F'F&F'!\"\"#F'\"\"**&,&F&\"\"#!#6F'F'F)!
\"#F,*&F&F'F)!\"$#F+\"\"$*&,&!\"%F'F&F'F',&F*F'F&F+F+#F+F-*&,&\"#8F'F&
F0F'F:F2F;*&,&F'F'F&F'F'F:F4#F'F6" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 10 "Exercice 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?T" }}
{PARA 15 "" 0 "" {TEXT -1 85 "The command with(orthopoly,T) allows the
 use of the abbreviated form of this command." }}{PARA 15 "" 0 "" 
{TEXT -1 88 "The function T computes the nth Chebyshev polynomial of t
he first kind, evaluated at x. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "T(2,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"TG6$\"\"#%\"XG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(orthopoly);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%\"GG%\"HG%\"LG%\"PG%\"TG%\"UG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "T(2,X);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*$%\"XG\"\"#F&!\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "seq(convert(1/T(n,X),parfrac,X),n=1..5);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6'*$%\"XG!\"\"*$,&*$F$\"\"#F)F%\"\"\"F%,&F##F%
\"\"$*&F$F*,&F(\"\"%!\"$F*F%#F0F-*$,(*$F$F0\"\")F(!\")F*F*F%,&F##F*\"
\"&*(F$F*,&!\"&F*F(F0F*,(F5\"#;F(!#?F:F*F%#!\"%F:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 256 "Ca ne peut rien donner de mieux...\nEn fait, Tn e
st de degr\351 n, coefficient dominant 2^(n-1), et v\351rifie Tn(cos(t
heta))=cos((n+1)theta) pour tout theta, donc Tn admet n racines simple
s (lesquelles pr\351cis\351ment ?). Il suffit ensuite de calculer des \+
r\351sidus..." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 4" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "On \351crit " }{XPPEDIT 18 0 "P=K*p
roduct((X-z[k])^(alpha[k]),k=1..n) " "/%\"PG*&%\"KG\"\"\"-%(productG6$
),&%\"XGF&&%\"zG6#%\"kG!\"\"&%&alphaG6#F0/F0;F&%\"nGF&" }{TEXT -1 21 "
 . F=P'/P vaut alors " }{XPPEDIT 18 0 "sum(alpha[k]/(X-z[k]),k=1..n" "
-%$sumG6$*&&%&alphaG6#%\"kG\"\"\",&%\"XGF*&%\"zG6#F)!\"\"F0/F);F*%\"nG
" }}{PARA 0 "" 0 "" {TEXT -1 141 "Consid\351rons maintenant une racine
 z de P' : si c'est une racine de P, c'est fini. Sinon, on peut \351va
luer F en z , ce qui donne la relation : " }{XPPEDIT 18 0 "sum(alpha[k
]/(z-z[k]),k=1..n" "-%$sumG6$*&&%&alphaG6#%\"kG\"\"\",&%\"zGF*&F,6#F)!
\"\"F//F);F*%\"nG" }{TEXT -1 18 "=0, soit encore : " }{XPPEDIT 18 0 "s
um(alpha[k]/(abs(z-z[k])^2)*(z-z[k]),k=1..n" "-%$sumG6$*(&%&alphaG6#%
\"kG\"\"\"*$-%$absG6#,&%\"zGF*&F06#F)!\"\"\"\"#F3,&F0F*&F06#F)F3F*/F);
F*%\"nG" }{TEXT -1 2 "=0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 11 "Ainsi, si a" }{XPPEDIT 18 0 "lambda[k]=alpha[k]/(a
bs(z-z[k])^2)" "/&%'lambdaG6#%\"kG*&&%&alphaG6#F&\"\"\"*$-%$absG6#,&%
\"zGF+&F16#F&!\"\"\"\"#F4" }{TEXT -1 35 " (qui est un r\351el >0), la \+
relation " }{XPPEDIT 18 0 "sum(lambda[k]*(z-z[k]),k=1..n" "-%$sumG6$*&
&%'lambdaG6#%\"kG\"\"\",&%\"zGF*&F,6#F)!\"\"F*/F);F*%\"nG" }{TEXT -1 
102 "=0 se traduit g\351om\351triquement par le fait que le point d'af
fixe z est barycentre des points d'affixes " }{XPPEDIT 18 0 "z[k]" "&%
\"zG6#%\"kG" }{TEXT -1 9 " : gagn\351." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 10 "Exercice 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "convert
((X^2+X+1)/((X^2-1)*(X^2+1)),parfrac,X);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(*$,&%\"XG\"\"\"!\"\"F'F(#\"\"$\"\"%*$,&F&F'F'F'F(#F(F+*&F&F',&
*$F&\"\"#F'F'F'F(#F(F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c
onvert(X/(X^4+X^2+1),parfrac,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
*$,(*$%\"XG\"\"#\"\"\"F'F)F)F)!\"\"#F*F(*$,(F&F)F'F*F)F)F*#F)F(" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 6" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 37 "convert(7/((X+1)^7-X^7-1),parfrac,X);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%\"XG!\"\"\"\"\"*$,&F%F'F'F'F&F&*$
,(*$F%\"\"#F'F%F'F'F'F&F&*$F+!\"#F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 36 "Vous vous attendiez \340 aussi simple ?" }}}}}{MARK "2 1 0 0" 
41 }{VIEWOPTS 1 1 0 1 1 1803 }