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}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "P
:=-7*x+5*x^2+75*x^3;Q:=x*(2*x-3*y)^2*(x-y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"PG,(%\"xG!\"(*&\"\"&\"\"\")F&\"\"#F*F**&\"#vF*)F&\"
\"$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG*(%\"xG\"\"\"),&F&\"
\"#*&\"\"$F'%\"yGF'!\"\"F*F',&F&F'F-F.F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "type(P,polynom),whattype(P),whattype(Q);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6%%%trueG%\"+G%\"*G" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 259 7 "nops(P)" }{TEXT -1 77 " don
ne le nombre de termes selon la forme , d\351velopp\351e ou factoris
\351e , de P :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "nops(P),nops(Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
\"\"$F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 5 "op(P)" }{TEXT -1 63 " \+
donne sous forme de s\351quence les termes ou les facteurs de P :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "op(
P);op(Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$%\"xG!\"(,$*$)F$\"\"#\"
\"\"\"\"&,$*$)F$\"\"$F*\"#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%\"xG*$
),&F#\"\"#*&\"\"$\"\"\"%\"yGF*!\"\"F'F*,&F#F*F+F," }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 261 33 "Coefficients , degr\351 , valuation:" }{TEXT 298 
34 "\nLes 2 syntaxes sont \351quivalentes:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "coeff(P,x^3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#
v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "coeff(P,x,3);  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#v" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 80 "Obtenir tous les coefficients de P (valable lorsque P est sous \+
forme d\351velopp\351e)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
coeffs(P);coeffs(Q); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&\"#v!\"(
" }}{PARA 8 "" 1 "" {TEXT -1 35 "Error, invalid arguments to coeffs\n
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "coeffs(expand(Q));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&!#;\"#@!\"*\"\"%" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 54 "Coefficient du terme de plus haut (de plus bas) de
gr\351:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lcoeff(P),tcoeff
(P);   " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#v!\"(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "lcoeff(Q,y),tcoeff(Q,y);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$,$%\"xG!\"*,$*$)F$\"\"%\"\"\"F)" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 19 "Degr\351 et valuation:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "degree(P),degree(Q,x),degree(Q,y); " }}{PARA 11 
"" 1 "" {XPPMATH 20 "6%\"\"$\"\"%F#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "ldegree(P),ldegree(Q,x),ldegree(Q,y); " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%\"\"\"F#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 262 28 "FONCTIONS SUR LES POLYNOMES:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 263 14 "quo(P,Q
,x,'r')" }{TEXT -1 80 " : quotient de la division euclidienne de P par
 Q , variable x , 'r' (optionnel)" }}{PARA 0 "" 0 "" {TEXT -1 41 "vari
able non assign\351e recevant le reste ." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "quo(x^5-1,x^2+x+1,x, 'r');r;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"F(*$)F&\"\"#F(!\"\"F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&!\"#\"\"\"%\"xG!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 264 14 "rem(P,Q,x,'q')" }{TEXT -1 77 " : reste de la divi
sion euclidienne de P par Q , variable x , 'q' (optionnel)" }}{PARA 0 
"" 0 "" {TEXT -1 70 "variable non assign\351e recevant le quotient .  \+
                        " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rem(x^5
-1,x^2+x+1,x, 'q');q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"#\"\"\"%
\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"F(*$
)F&\"\"#F(!\"\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT 265 13 "Discriminant:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "discrim(x^3+a*x+b,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%
\"aG\"\"$\"\"\"!\"%*&\"#FF()%\"bG\"\"#F(!\"\"" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 266 43 "Evaluer, ordonner, transformer un polyn\364me:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sub
s(x=1+2*I,y=I,Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$!#:\"\"&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Ordonner Q selon les puissances d
\351croissantes de y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "so
rt(expand(Q),y);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%\"xG\"\"\")
%\"yG\"\"$F&!\"**(\"#@F&)F%\"\"#F&)F(F.F&F&*(\"#;F&)F%F)F&F(F&!\"\"*&
\"\"%F&)F%F5F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "D\351velopper
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "expand((x-3*y+a)*(1-a^
2+x+y));  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8%\"xG\"\"\"*&F$F%%\"aG
\"\"#!\"\"*$F$F(F%*&F$F%%\"yGF%!\"#F,!\"$*&F,F%F'F(\"\"$*$F,F(F.F'F%*$
F'F0F)*&F'F%F$F%F%*&F'F%F,F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 
"Regrouper les mon\364mes de l'expression en l'ind\351termin\351e a :
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "collect((x-3*y+a)*(1-a^
2+x+y),a);   " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"aG\"\"$\"\"\"
!\"\"*&,&%\"xGF)*&F'F(%\"yGF(F(F()F&\"\"#F(F(*&,(F(F(F,F(F.F(F(F&F(F(*
&,&F,F(*&F'F(F.F(F)F(F2F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Me
ttre P sous la forme de H\366rner:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "P;H:=convert(P,horner);  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(%\"xG!\"(*&\"\"&\"\"\")F$\"\"#F(F(*&\"#vF()F$\"\"$F(F
(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG*&,&!\"(\"\"\"*&,&\"\"&F(*&
\"#vF(%\"xGF(F(F(F.F(F(F(F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "with(codegen,cost);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%%costG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Ce qui est moins co\373teux e
n op\351rations:" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cost(P),cost(H);  " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$,&%*additionsG\"\"#*&\"\"'\"\"\"%0multiplicationsG
F(F(,&F)\"\"$*&F%F(F$F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Fact
oriser sur le corps Q des rationnels" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "factor(x^3+x^2+x+1); " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&,&\"\"\"F%%\"xGF%F%,&*$)F&\"\"#F%F%F%F%F%" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 11 "Normaliser:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "normal((4*x-3)^2+x-1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*
$)%\"xG\"\"#\"\"\"\"#;*&\"#BF(F&F(!\"\"\"\")F(" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
267 39 "RACINES ET FACTORISATION DES POLYN\324MES:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Valeur approch\351
es des z\351ros r\351els ou complexes avec " }{TEXT 268 6 "fsolve" }
{TEXT -1 2 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "P:=x^3+x+1:fsolve
(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+Q!yK#o!#5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(P,x,complex);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%$!+Q!yK#o!#5^$$\"+>!R;T$F%$!++9ah6!\"*^$F'$\"++9ah6F+" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 
"Localiser les z\351ros r\351els dans un intervalle de longueur voulue
 , ici 10^-3 , avec " }{TEXT 269 8 "realroot" }{TEXT -1 2 " :" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "realroot(P,0.001);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7#7$#!$*p\"%C5#!$\\$\"$7&" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 22 "Calcul des z\351ros avec " }{TEXT 270 9 "solve(P) " }
{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(x^4+1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&,&*$-%%sqrtG6#\"\"#\"\"\"#F)F(*&^#F*F)
F%F)F),&F+F)*&#F)F(F)F$F)!\"\",&F$#F0F(*&^#F2F)F%F)F),&F3F)*&F*F)F%F)F
)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
16 "Par la fonction " }{TEXT 271 6 "factor" }{TEXT -1 25 ", on peut fa
ctoriser sur " }{TEXT 296 1 "Q" }{TEXT -1 46 " ou sur le corps induit \+
par les coefficients :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "factor(x^
4-1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"\"F&!\"\"F&,&F%
F&F&F&F&,&*$)F%\"\"#F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "factor(x^3-I*x-I+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&
%\"xG\"\"\"^#F'F'F',&F&!\"\"^$F'F'F'F',&F&F'F'F'F'F*" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 272 39 "Pour obtenir l
a factorisation compl\350te:" }}{PARA 0 "" 0 "" {TEXT 299 7 " factor" 
}{TEXT 303 4 "(P,\{" }{XPPEDIT 18 0 "alpha[1],alpha[2];" "6$&%&alphaG6
#\"\"\"&F$6#\"\"#" }{TEXT 300 5 ",...," }{XPPEDIT 18 0 "alpha[n];" "6#
&%&alphaG6#%\"nG" }{TEXT 301 2 "\})" }{TEXT 302 2 "  " }{TEXT -1 77 "a
utorise Maple \340 factoriser P sur un corps contenant les nombres alg
\351briques\n" }{XPPEDIT 18 0 "alpha[1],alpha[2];" "6$&%&alphaG6#\"\"
\"&F$6#\"\"#" }{TEXT 304 5 ",...," }{XPPEDIT 18 0 "alpha[n];" "6#&%&al
phaG6#%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "factor(x^4-1,I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$**,&%\"xG\"\"\"^#F'F'F',&F&!\"\"F(F'F',&F&F'F'F'F',&F&F'F'F*F'F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(x^3-x^2+2*x-2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"F&!\"\"F&,&*$)F%\"\"#F&
F&F+F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "factor(x^3-x^2+
2*x-2,\{I,sqrt(2)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"
\"*&^#!\"\"F&-%%sqrtG6#\"\"#F&F&F&,&F%F&*&F*F&^#F&F&F&F&,&F%F&F&F)F&" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 
"On peut aussi utiliser " }{TEXT 273 5 "Split" }{TEXT -1 41 " , qui do
nne une factorisation compl\350te :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "with(PolynomialTools,Split):P:=Split(x^4-1,x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"PG**,&%\"xG\"\"\"-%'RootOfG6#,&*$)%#_ZG\"\"#F(F(F
(F(F(F(,&F'F(F(!\"\"F(,&F'F(F)F2F(,&F'F(F(F(F(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 274 6 "RootOf" }{TEXT -1 77 " d\351signe ici toute racine d
u polyn\364me _Z\262+1, on peut simplifier l'\351criture en" }}{PARA 
0 "" 0 "" {TEXT -1 10 "utilisant " }{TEXT 275 5 "alias" }{TEXT -1 1 ":
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "alias(alpha=RootOf(_Z^2+1)):P;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\"%&alphaGF&F&,&F%F&F
&!\"\"F&,&F%F&F'F)F&,&F%F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
46 "Pour calculer les valeurs possibles de alpha :" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 28 "allvalues(RootOf(_Z^2+1));  " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$^#\"\"\"^#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "La fonction " }{TEXT 276 5 "ro
ots" }{TEXT -1 50 " donne les z\351ros avec leur ordre de multiplicit
\351:\n" }{XPPEDIT 18 0 "x^2+1" "6#,&*$%\"xG\"\"#\"\"\"F'F'" }{TEXT 
-1 22 " n'a aucun z\351ro sur Q:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"roots(x^2+1);      " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "On calcule ses z\351ros sur le cor
ps Q(I) induit par les coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "roots(x^2+1,I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$
7$^#!\"\"\"\"\"7$^#F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 277 33 "FRACTIONS RATIONNELLES : de type " }{TEXT 278 7 "
ratpoly" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "F:=(2*x^2+x-3)^2/(x^4-1);numer(F),denom(F);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"FG*&,(*$)%\"xG\"\"#\"\"\"F*F)F+\"\"$!\"
\"F*,&*$)F)\"\"%F+F+F+F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$),(*$)
%\"xG\"\"#\"\"\"F)F(F*\"\"$!\"\"F)F*,&*$)F(\"\"%F*F*F*F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(F);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*(,&%\"xG\"\"#\"\"$\"\"\"F(,(*$)F%F&F(F&F%F(F'!\"\"F(,*
*$)F%F'F(F(F*F(F%F(F(F(F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 17 "Fac
torisation sur" }{TEXT 305 1 " " }{TEXT 280 8 "IR ou C:" }}{PARA 0 "" 
0 "" {TEXT -1 21 "factorisation sur IR:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "factor(F);   " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"
#\"\"$\"\"\"F&,&F%F(!\"\"F(F(,&F%F(F(F(F*,&*$F%F&F(F(F(F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "factorisation sur C:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "factor(F,I); " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*,,&%\"xG\"\"\"F&!\"\"F&,&F%\"\"#\"\"$F&F),&F%F&^#F&F&F',&F%F&^#
F'F&F',&F%F&F&F&F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "retour \340 la f
orme normale:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "normal(%,e
xpanded); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**$)%\"xG\"\"$\"\"\"
\"\"%*&\"\")F))F'\"\"#F)F)*&F(F)F'F)!\"\"\"\"*F0F),*F%F)*$F-F)F)F'F)F)
F)F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 281 46 "D\351composition en \351l\351ments s
imples sur IR ou C:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"" 0 "" {TEXT 306 18 "convert(F,parfrac," }{TEXT 308 1 "x" }{TEXT 309 
2 ") " }{TEXT -1 74 "d\351compose en \351l\351ments simples la fractio
n rationnelle F en l'ind\351termin\351e " }{TEXT 307 1 "x" }{TEXT -1 
2 ".\n" }{TEXT 310 18 "convert(F,parfrac," }{TEXT 311 2 "x," }{TEXT 
315 1 "\{" }{XPPEDIT 18 0 "alpha[1],alpha[2];" "6$&%&alphaG6#\"\"\"&F$
6#\"\"#" }{TEXT 313 5 ",...," }{XPPEDIT 18 0 "alpha[n];" "6#&%&alphaG6
#%\"nG" }{TEXT 314 1 "\}" }{TEXT 312 2 ") " }{TEXT -1 76 "autorise Map
le \340 d\351composer sur un corps contenant les\n nombres alg\351briq
ues " }{XPPEDIT 18 0 "alpha[1],alpha[2];" "6$&%&alphaG6#\"\"\"&F$6#\"
\"#" }{TEXT 316 5 ",...," }{XPPEDIT 18 0 "alpha[n];" "6#&%&alphaG6#%\"
nG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 21 "D\351composition sur IR:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "convert(F,parfrac,x);    " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
(\"\"%\"\"\"*&F%F%,&%\"xGF%F%F%!\"\"F)*&,&!#7F%*&\"\"&F%F(F%F%F%,&*$)F
(\"\"#F%F%F%F%F)F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "D\351compos
ition sur C:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(F,p
arfrac,x,I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"%\"\"\"*&^$#\"\"
&\"\"#!\"'F%,&%\"xGF%^#F%F%!\"\"F%*&^$#!\"&F*F+F%,&F-F/F.F%F/F%*&F%F%,
&F-F%F%F%F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 19 "Exercice corrig\351 5:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "" 0 "" {TEXT 284 6 "Ex 5.1" }{TEXT -1 37 ": D\351comp
oser en \351l\351ments simples sur " }{TEXT 282 2 "IR" }{TEXT 317 1 " \+
" }{TEXT -1 11 ", puis sur " }{TEXT 283 1 "C" }{TEXT -1 15 " , la frac
tion " }{XPPEDIT 18 0 "F:=1/(x^4+1)" "6#>%\"FG*&\"\"\"F&,&*$%\"xG\"\"%
F&F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "F:=1/(x^4+1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG*&\"\"\"F&,&*$)%\"xG\"\"%F&
F&F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "convert(F,pa
rfrac,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"xG\"\"
%F$F$F$F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "F n'est pas d
\351compos\351e. Voyons les racines du d\351nominateur: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "\{solve(denom(F))\}; " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<&,&*$-%%sqrtG6#\"\"#\"\"\"#F*F)*&^#F+F*F&F*F*,&
F%#!\"\"F)*&^#F/F*F&F*F*,&F,F**&#F*F)F*F%F*F0,&F1F**&F+F*F&F*F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "On obtient la d\351composition sur
 IR par:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "convert(F,parfr
ac,x,sqrt(2)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&!\"#\"\"\"*&%
\"xGF'-%%sqrtG6#\"\"#F'F'F',(*$)F)F-F'F'F(!\"\"F'F'F1#F1\"\"%*(#F'F3F'
,&F-F'F(F'F',(F/F'F(F'F'F'F1F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 
"On obtient la d\351composition sur C par:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "convert(F,parfrac,x,\{I,sqrt(2)\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,**(^$#!\"\"\"\"%#\"\"\"F(F*\"\"##F*F+,(%\"xGF+*$-
%%sqrtG6#F+F*F'*&F0F*^#F*F*F*F'F**(^$F&F&F*F+F,,(F.F+F/F'*&^#F'F*F0F*F
*F'F**(^$F)F&F*F+F,,(F.F+F/F*F8F*F'F**(^$F)F)F*F+F,,(F.F+F/F*F3F*F'F*
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 285 6 "Ex 5.2" }{TEXT -1 7 ": Soit " }
{XPPEDIT 18 0 "P = X (X-1) (X-2) (X-3)" "6#/%\"PG---%\"XG6#,&F(\"\"\"F
+!\"\"6#,&F(F+\"\"#F,6#,&F(F+\"\"$F," }{TEXT -1 2 " ." }}{PARA 0 "" 0 
"" {TEXT -1 41 "Montrer que les z\351ros du polyn\364me d\351riv\351 \+
" }{TEXT 286 2 "P'" }{TEXT -1 52 " sont 3 termes cons\351cutifs d'une \+
suite arithm\351tique." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "P:=X*(X-1)*(X-2)*(X-3):dP:=diff(exp
and(P),X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dPG,**$)%\"XG\"\"$\"
\"\"\"\"%*&\"#=F*)F(\"\"#F*!\"\"*&\"#AF*F(F*F*\"\"'F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "zeros:=[solve(dP)];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&zerosG7%#\"\"$\"\"#,&*$-%%sqrtG6#\"\"&\"\"\"#F/
F(F&F/,&F&F/*&#F/F(F/F*F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 
"On trie les \351l\351ments de la liste par ordre croissant ," }{TEXT 
329 5 " sort" }{TEXT -1 26 " ne fonctionnant pas ici :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "tri:=proc(L::list)\n local x,y,Z;
\n Z:=L;\n for x to nops(Z)-1 do\n   for y from x+1 to nops(Z) do\n   \+
 if evalf(Z[y]-Z[x])<0 then Z:=subsop(x=Z[y],y=Z[x],Z) end if;\n   end
 do;\n end do;\n Z;\n end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "zeros:=tri(zeros);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&zeros
G7%,&#\"\"$\"\"#\"\"\"*&#F*F)F**$-%%sqrtG6#\"\"&F*F*!\"\"F',&F-#F*F)F'
F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "On calcule les \351carts en
tre 2 termes cons\351cutifs , la raison est " }{XPPEDIT 18 0 "sqrt(5)/
2 " "6#*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 1 ":" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 61 "for k to nops(zeros)-1 do print(zeros[k+1]-ze
ros[k]) end do; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"
&\"\"\"#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"&
\"\"\"#F)\"\"#" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT -1 17 "Travail dirig\351 5:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "" 0 "" {TEXT 318 7 "TD 5.1:" }{TEXT -1 30 "\nSoit la \+
fraction rationnelle " }{TEXT 287 2 "F " }{TEXT -1 14 "d\351finie par \+
: " }{XPPMATH 20 "6#/%\"FG*(%\"XG\"\"',&*$F&\"\"#\"\"\"F+F+!\"#,&F&F+F
+F+F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "
1\260 La d\351composition de " }{TEXT 288 1 "F" }{TEXT -1 26 " en \351
l\351ments simples dans " }{TEXT 289 5 "IR(X)" }{TEXT -1 10 " s'\351cr
it :" }}{PARA 257 "" 1 "" {XPPMATH 20 "6#>%\"GG,,%\"aG\"\"\"*&,&*&%\"b
GF'%\"XGF'F'%\"cGF'F',&*$F,\"\"#F'F'F'!\"\"F'*&,&*&%\"dGF'F,F'F'%\"eGF
'F'F.!\"#F'*&%\"fGF',&F,F'F'F'F1F'*&%\"gGF'F:F7F'" }}{PARA 0 "" 0 "" 
{TEXT -1 37 "   On r\351duit les fractions composant " }{TEXT 290 1 "G
" }{TEXT -1 41 " au m\352me d\351nominateur et on identifie les" }}
{PARA 0 "" 0 "" {TEXT -1 19 "    num\351rateurs de " }{TEXT 291 1 "F" 
}{TEXT -1 7 " et de " }{TEXT 292 1 "G" }{TEXT -1 28 ". En d\351duire l
es valeurs de " }{TEXT 293 13 "a,b,c,d,e,f,g" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "2\260 Cal
culer la d\351composition de " }{TEXT 294 1 "F" }{TEXT -1 26 " en \351
l\351ments simples dans " }{TEXT 295 4 "C(X)" }{TEXT -1 2 " ." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 321 7 "TD 5.2:" }{TEXT -1 25 "\nSoit P un
 polyn\364me de K[" }{TEXT 327 1 "x" }{TEXT -1 35 "] , s'\351crivant s
ous la forme :     " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6#%\"xG-
%$SumG6$*&&%\"aG6#%\"kG\"\"\")F'F/F0/F/;\"\"!%\"nG" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "L'algorithme de H\366rner
 permet d'\351crire P(" }{TEXT 322 1 "x" }{TEXT -1 17 ") sous la forme
 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"PG6#%\"xG,&*&,&*&,(*&,&*&,&*&&%\"aG6#%\"nG\"\"\"F'F6F6&F36#,&F5F6
!\"\"F6F6F6F'F6F6&F36#,&F5F6!\"#F6F6F6F'F6F6%(~.~.~.~GF6&F36#\"\"#F6F6
F'F6F6&F36#F6F6F6F'F6F6&F36#\"\"!F6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 319 8 "Exemple:" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"QG,,*$%\"xG\"\"%\"\"$*$F'F)\"\"#*$F'F+!\"\"F'F+\"\"\"F." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG,&*&,&*&,&*&,&%\"xG\"\"$\"\"#\"
\"\"F/F,F/F/!\"\"F/F/F,F/F/F.F/F/F,F/F/F/F/" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Ecrire une proc\351dure PolyHor
ner(P , " }{TEXT 324 1 "x" }{TEXT -1 25 ")  permettant d'\351crire P(
" }{TEXT 323 1 "x" }{TEXT -1 21 ") sous cette forme , " }}{PARA 0 "" 
0 "" {TEXT -1 31 "P \351tant un polyn\364me donn\351 de K[" }{TEXT 
326 1 "x" }{TEXT -1 4 "] , " }{TEXT 325 1 "x" }{TEXT -1 30 " \351tant \+
le nom de la variable ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 44 "NB:  il est interdit d'utiliser la fonction " }
{TEXT 320 6 "horner" }{TEXT -1 11 " de MAPLE ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Grace \340 la fonction " 
}{TEXT 330 4 "cost" }{TEXT -1 76 " , \351valuer le co\373t en op\351ra
tions n\351cessaires pour \351valuer Q et\nPolyHorner(Q," }{TEXT 328 
1 "x" }{TEXT -1 42 ") , Q \351tant le polyn\364me donn\351 en exemple \+
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "2 0" 39 }{VIEWOPTS 1 1 
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }