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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 30 "Feuille d'accompagnement
 TD : " }}{PARA 257 "" 0 "" {TEXT -1 29 "alg\350bre lin\351aire et pol
ynomes" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 9" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P:=n->(X+1)^n-(X-1)^n:P(3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$,&%\"XG\"\"\"F'F'\"\"$F'*$,&F&F'!
\"\"F'F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"XG\"\"#\"\"'F&\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "degree(\",X),lcoeff(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "degree(expand(P(1515)),X),lcoeff(expand(P(1515)));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%9:\"%II" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "degree(expand(P(1789)),X),lcoeff(expand(P(1789
)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%)y\"\"%yN" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "Q:=n->(X^2+1)^n-(X^2-1)^n:Q(1515);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$,&*$%\"XG\"\"#\"\"\"F)F)\"%::F)*$,
&F&F)!\"\"F)F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "degree(
expand(Q(1515)),X),lcoeff(expand(Q(1515)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"%GI\"%II" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "degree(expand(Q(1789)),X),lcoeff(expand(Q(1789)));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$\"%wN\"%yN" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
11 "Exercice 10" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "horner:=
proc(l,x)\nlocal res,n,k;\n n:=nops(l)-1; res:=l[n+1]; # res<-an\n for
 k from 1 to n do  #res<-res*x+a[n-k]\n     res:=res*x+l[n-k+1] od;\n \+
RETURN(res)\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'hornerGf*6$%\"
lG%\"xG6%%$resG%\"nG%\"kG6\"F-C&>8%,&-%%nopsG6#9$\"\"\"!\"\"F6>8$&F56#
,&F0F6F6F6?(8&F6F6F0%%trueG>F9,&*&F9F69%F6F6&F56#,(F0F6F>F7F6F6F6-%'RE
TURNG6#F9F-F-6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "horner(
[1,3,-2,3,0,1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*&,&*$
%\"xG\"\"#\"\"\"\"\"$F-F-F+F-F-!\"#F-F-F+F-F-F.F-F-F+F-F-F-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,,*$%\"xG\"\"&\"\"\"*$F%\"\"$F)*$F%\"\"#!\"#F%F)F'
F'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 12" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(orthopoly);" }}{PARA 7 "" 1 "
" {TEXT -1 29 "Warning, new definition for P" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7(%\"GG%\"HG%\"LG%\"PG%\"TG%\"UG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "seq(T(k,X),k=0..5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6(\"\"\"%\"XG,&*$F$\"\"#F'!\"\"F#,&*$F$\"\"$\"\"%F$!\"$,(
*$F$F,\"\")F&!\")F#F#,(*$F$\"\"&\"#;F*!#?F$F4" }}}{EXCHG {PARA 15 "" 
0 "" {TEXT -1 87 "The function T computes the nth Chebyshev polynomial
 of the first kind, evaluated at x." }}{PARA 15 "" 0 "" {TEXT -1 49 "T
hey satisfy the following recurrence relation: \n" }{TEXT 23 91 "     \+
   T(0,x) = 1,\n        T(1,x) = x,\n        T(n,x) = 2*x*T(n-1,x) - T
(n-2,x),  for n>1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "seq(U(k,X),k=0..5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6(\"\"\",$%\"XG\"\"#,&*$F%F&\"\"%!\"\"F#,&*$F%\"\"$\"\")F
%!\"%,(*$F%F)\"#;F(!#7F#F#,(*$F%\"\"&\"#KF,!#KF%\"\"'" }}}{EXCHG 
{PARA 15 "" 0 "" {TEXT -1 93 "The function U will compute the nth Cheb
yshev polynomial of the second kind, evaluated at x. " }}{PARA 15 "" 
0 "" {TEXT -1 49 "They satisfy the following recurrence relation: \n" 
}{TEXT 23 93 "        U(0,x) = 1,\n        U(1,x) = 2*x,\n        U(n,
x) = 2*x*U(n-1,x) - U(n-2,x),  for n>1." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(X=cos(theta),T(10
,X));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$-%$cosG6#%&thetaG\"#5\"$7
&*$F%\"\")!%!G\"*$F%\"\"'\"%?6*$F%\"\"%!$+%*$F%\"\"#\"#]!\"\"\"\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "combine(\",trig);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#,$%&thetaG\"#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(X=cos(theta),U(10,X));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$-%$cosG6#%&thetaG\"#5\"%C5*$F%\"\"
)!%/B*$F%\"\"'\"%#z\"*$F%\"\"%!$g&*$F%\"\"#\"#g!\"\"\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "combine(\"*sin(theta),trig);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,$%&thetaG\"#6" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 18 "Exercices 13 et 14" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "A:=X^6-4*X^3+2*X^2-1;B:=X^2+4;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG,**$%\"XG\"\"'\"\"\"*$F'\"\"$!\"%*$F'\"\"#F.
!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,&*$%\"XG\"\"#\"\"\"
\"\"%F)" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 200 "The rem function ret
urns the remainder of a divided by b. The quo function returns the quo
tient of a divided by b. The remainder r and quotient q satisfy: a = b
*q + r where degree(r,x) < degree(b,x). " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "quo(A,B,X),rem(A,B,X);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$,**$%\"XG\"\"%\"\"\"*$F%\"\"#!\"%F%F*\"#=F',&!#tF'F%\"#;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "C:=4*X^3+X^2:D:=X+1+I:" }}
{PARA 8 "" 1 "" {TEXT -1 53 "Error, attempting to assign to `D` which \+
is protected" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "D d\351signe l'op
\351rateur de d\351rivation..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "C:=4*X^3+X^2:E:=X+1+I:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "quo(C,E,X),rem(C,E,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,*!
\"\"\"\"\"%\"IG\"\"(*&,&!\"$F%F&!\"%F%%\"XGF%F%*$F,\"\"#\"\"%,&\"\")F%
F&!\"'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 23" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "P:=n->X^(2*n+1)-(2*n+1)*X^(n
+1)+(2*n+1)*X^n-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGf*6#%\"nG6
\"6$%)operatorG%&arrowGF(,*)%\"XG,&9$\"\"#\"\"\"F2F2*&F/F2)F.,&F0F2F2F
2F2!\"\"*&F/F2)F.F0F2F2F6F2F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "subs(X=1,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(X=1,diff(P(n),X
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"nG\"\"#\"\"\"F&*&,&F$F%F&F
&F&,&F$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 33 "expand(subs(X=1,diff(P(
n),X$2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "expand(subs(X=1,diff(P(n),X$3)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"nG\"\"\"*$F$\"\"#\"\"$*$F$F(F'" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Q:=n->X^(2*n)-n^2*X^(n+1)+
2*(n^2-1)*X^n-n^2*X^(n-1)+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG
f*6#%\"nG6\"6$%)operatorG%&arrowGF(,,)%\"XG,$9$\"\"#\"\"\"*&F0F1)F.,&F
0F2F2F2F2!\"\"*&,&*$F0F1F2F6F2F2)F.F0F2F1*&F0F1)F.,&F0F2F6F2F2F6F2F2F(
F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(X=1,Q(n));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "expand(subs(X=1,diff(Q(n),X)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "exp
and(subs(X=1,diff(Q(n),X$2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "expand(subs(X=1,diff(Q
(n),X$3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "expand(subs(X=1,diff(Q(n),X$4)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"nG\"\"%\"\"#*$F%F'!\"#" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 25" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "P:=16*X^5-20*X^3+5*X-1;factor(P);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"PG,**$%\"XG\"\"&\"#;*$F'\"\"$!#?F'F(!\"\"\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"XG\"\"\"!\"\"F&F&,(F'F&
F%\"\"#*$F%F)\"\"%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solv
e(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\",&*$\"\"&#F#\"\"##F#\"
\"%#!\"\"F*F#,&F+F#F%F+F$F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "On
 va dire  \340 Maple de travailler sur un corps plus riche que Q, \340
 savoir Q[sqrt(5)]..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fa
ctor(P,\{sqrt(5)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"XG\"
\"\"!\"\"F'F',(F&\"\"%F'F'*$\"\"&#F'\"\"#F'F.,(F&F*F'F'F+F(F.#F'\"#;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(P,complex);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"XG\"\"\"$!\"\"\"\"!F'F',&F&F'
$\"+W*p,4)!#5F'\"\"#,&F&F'$!+W*p,4$F.F'F/$\"#;F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 41 "Il s'agit d'une factorisation \"num\351rique\"" }}
}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 26" }}{SECT 0 {PARA 4 
"" 0 "" {TEXT -1 5 "X^8-1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"P:=X^8-1:factor(P);" }}{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 "" {TEXT -1 43 "Il s'agit en fait de la factorisat
ion sur Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(P);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\"\"!\"\"%\"IG,$F%F$,&*$\"\"##F#F)F*
*&F%F#F)F*F*,&F(#F$F)F+F-,&F(F*F+F-,&F(F-F+F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "factor(P,\{sqrt(2)\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*,,&*$%\"XG\"\"#\"\"\"F(F(F(,(F%F(*&F'#F(F'F&F(F(F(F(F(
,(F%F(F*!\"\"F(F(F(,&F&F(F-F(F(,&F&F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "factor(P,\{sqrt(2),I\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*2,&%\"XG\"\"\"%\"IGF'F',&F&F'F(!\"\"F',(F&\"\"#*$F,#
F'F,F**&F(F'F,F.F'F',(F&F,F-F'F/F*F',(F&F,F-F*F/F*F',(F&F,F-F'F/F'F',&
F&F'F*F'F',&F&F'F'F'F'#F'\"#;" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 
"X^8+X^4+1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "P:=X^8+X^4+1:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(P);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#*(,(*$%\"XG\"\"#\"\"\"F&F(F(F(F(,(F%F(F&!\"\"F(
F(F(,(*$F&\"\"%F(F%F*F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "solve(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*,&#!\"\"\"\"#\"\"\"*&%
\"IGF'\"\"$#F'F&F+,&F$F'F(F$,&F+F'F(F$,&F+F'F(F+,$*$,&F&F'F(!\"#F+F+,$
F0F$,$*$,&F&F'F(F&F+F+,$F5F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "P
ouah !!!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalc(\{\"\});
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*,&#\"\"\"\"\"#F&*&%\"IGF&\"\"$F%
#!\"\"F',&F+F&F(F+,&F%F&F(F%,&F+F&F(F%,&*$F*F%F%F)F+,&F1F%F)F%,&F1F+F)
F+,&F1F+F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(P,sq
rt(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,(*$%\"XG\"\"#\"\"\"*&\"
\"$#F(F'F&F(F(F(F(F(,(F%F(F)!\"\"F(F(F(,(F%F(F&F-F(F(F(,(F%F(F&F(F(F(F
(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(P,\{sqrt(3),I\}
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*2,(%\"XG\"\"#*$\"\"$#\"\"\"F'
!\"\"%\"IGF+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+F)F*F,F+,(F&F'F,F+F2F+F+,(F&F'F+F+F2F+F+,(F&F'F+F+F2F,F+#F+\"$
c#" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 30" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "P:=X^3-4*X^2+2*X-1;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"PG,**$%\"XG\"\"$\"\"\"*$F'\"\"#!\"%F'F+!\"\"F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sols:=\{solve(P)\};" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG<%,(*$,&\"$K$\"\"\"*$\"$@$#F*
\"\"#\"#7#F*\"\"$#F*\"\"'*$F(#!\"\"F1#\"#?F1#\"\"%F1F*,*F'#F6F/F4#!#5F
1F9F**(%\"IGF*F1F-,&F'F2F4#!#?F1F*#F6F.,*F'F<F4F=F9F*F?F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sols[1]^2+sols[2]^2+sols[3]^2;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*$,(*$,&\"$K$\"\"\"*$\"$@$#F)\"\"#\"
#7#F)\"\"$#F)\"\"'*$F'#!\"\"F0#\"#?F0#\"\"%F0F)F-F)*$,*F&#F5F.F3#!#5F0
F8F)*(%\"IGF)F0F,,&F&F1F3#!#?F0F)#F5F-F-F)*$,*F&F<F3F=F8F)F?F,F-F)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"sols[1]^3+sols[2]^3+sols[3]^3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
21 "R\351sultat illisible..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "expand(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$,&\"$K$\"\"\"
*$\"$@$#F'\"\"#\"#7!\"\"#\"%+!)\"\"*#\"$\"p\"#=F'F(#F'\"\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"#V" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "NB
 : lorsque Maple ne sait pas trouver explicitement les racines d'un po
lyn\364me, il peut cependant \351valuer des expressions sym\351triques
 en ces racines, gr\342ce  RootOf, mais \347a devient plus ardu..." }}
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