{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 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 "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 11 "CHAPITRE II" }{TEXT 299 58 " : Nombres complexes, arithm\351tique,\npolyn\364mes et \351qu ations" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 266 10 "Exercice 4" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 260 1 "a" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 4 "z||1" }{TEXT -1 65 " est une indexation en Maple. Cela co rrespond math\351matiquement \340 " }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6# \"\"\"" }{TEXT -1 86 ". Cette forme est utilisable dans des boucles (p ar exemple), sous sa forme \"g\351n\351rale\" " }{TEXT 258 4 "z||i" } {TEXT -1 73 ". C'est assez pratique, pour d\351finir des suites de nom bres par exemple..." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "z||1 := 6+I*Pi;\nconvert(z||1, polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G,&\"\"'\"\"\"*&^#F'F'%#PiGF'F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&polarG6$*$-%%sqrtG6#,&\"#O\"\"\"*$)%#PiG\"\"#F,F,F ,-%'arctanG6#,$F/#F,\"\"'" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Que stion " }{TEXT 259 1 "b" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "z ||2 := (3-6*I)^(3*I);\nRe(z||2); Im(z||2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z2G)^$\"\"$!\"'^#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$-%'arctanG6#\"\"#\"\"$\"\"\"-%$cosG6#,$-%#lnG6#\"#X#F, F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$-%'arctanG6#\"\"# \"\"$\"\"\"-%$sinG6#,$-%#lnG6#\"#X#F,F+F-" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 9 "Question " }{TEXT 261 1 "c" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "z||3 := I-1; z||4 := 2*I;\nabs(z||3); argument(z||3); \nabs(z||4); argument(z||4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z3G ^$!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z4G^#\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"#" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Une variante consiste \340 conve rtir le nombre complexe de la forme alg\351brique vers la forme polair e, ce qui donne directement le module et l'argument !" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "convert(z||3, polar); conver t(z||4, polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$*$-%%sqr tG6#\"\"#\"\"\",$%#PiG#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %&polarG6$\"\"#,$%#PiG#\"\"\"F&" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 264 1 "d" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "P our la forme alg\351brique, Maple simplifie automatiquement vers une f orme g\351n\351rale " }{TEXT 263 1 "a" }{TEXT -1 3 " + " }{TEXT 262 2 "ib" }{TEXT -1 1 "." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z||3/z||4;\nconvert(%, polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$#\"\"\"\"\"#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$, $*$-%%sqrtG6#\"\"#\"\"\"#F,F+,$%#PiG#F,\"\"%" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 265 1 "e" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "abs(z||2); argument(z||2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$absG6#)^$\"\"$!\"'^#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"#X#\"\"$\"\"#*&F*\"\"\"%#PiGF,!\"\"" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 10 "Exercice 5" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 276 1 "a" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "a:=1049427; b:=17493;\nigcd(a,b); isprime(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"(F%\\5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"b G\"&$\\<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Quest ion " }{TEXT 277 1 "b" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ifa ctor(a); ifactor(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()-%!G6#\"\"$ \"\"#\"\"\"-F&6#\"#F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*(-%!G6#\"\"$\"\"\")-F%6#\"\"(F'F(-F%6#\"#F'" }{TEXT -1 12 ". En effet, " } {TEXT 267 1 "a" }{TEXT -1 21 " peut aussi s'\351crire " }{XPPEDIT 18 0 "``(3)^2*``(7)^`0`*``(17)*``(19)^3;" "6#**-%!G6#\"\"$\"\"#)-F%6#\"\" (%\"0G\"\"\"-F%6#\"#F'" }{TEXT -1 4 " et " }{TEXT 268 1 "b " }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(3)*``(7)^3*``(17)*``(19)^`0`;" "6#**-%!G6#\"\"$\"\"\"*$-F%6#\"\"(F'F(-F%6#\"#%\"0GF(" } {TEXT -1 112 ". On prend ensuite chaque facteur (3, 7, 17 et 19), que \+ l'on affecte de l'exposant le plus \351lev\351 rencontr\351 dans " } {TEXT 269 1 "a" }{TEXT -1 4 " ou " }{TEXT 270 1 "b" }{TEXT -1 67 " : a insi, le facteur 3 sra affect\351 de l'exposant 2 (rencontr\351 dans \+ " }{TEXT 271 1 "a" }{TEXT -1 42 "), 7 aura pour exposant 3 (rencontr \351 dans " }{TEXT 272 1 "b" }{TEXT -1 40 "), 17 aura l'exposant 1 (re ncontr\351 dans " }{TEXT 273 1 "a" }{TEXT -1 4 " ou " }{TEXT 274 1 "b " }{TEXT -1 37 ") et 19 l'exposant 3 (rencontr\351 dans " }{TEXT 275 1 "a" }{TEXT -1 94 "). On se r\351f\350rera au cours de DEUG MIAS 1 \350re ann\351e pour la d\351monstration th\351orique de ceci... " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "On v\351rifie tout de m\352me notr e r\351sultat :" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ifactor(ilcm(a,b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**)-%!G6#\" \"$\"\"#\"\"\")-F&6#\"\"(F(F*-F&6#\"#F(F*" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 280 1 "c" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "773^3570 mod 3571;\n123^3570 mod 35 71;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Si n est un nombre entier compris entre 1 et 3570, alors le r\351sultat de " } {XPPEDIT 18 0 "n^3570;" "6#*$%\"nG\"%qN" }{TEXT -1 166 " mod 3571 doit \352tre 1. C'est le petit th\351or\350me de Fermat qui l'affirme. En \+ effet, il faut que 3571 soit premier (ce qui est le cas, vu l'instruct ion suivante) et que " }{TEXT 278 1 "n" }{TEXT -1 57 " ne soit pas div isible par 3571 (on est dans ce cas, car " }{TEXT 279 1 "n" }{TEXT -1 30 " est compris entre 1 et 3570)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "isprime(3571);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%t rueG" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 281 1 "d" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ithprime(500);\nnextpr ime(40000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%rN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&4+%" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Ques tion " }{TEXT 283 1 "e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "if actor(200!);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*hp)-%!G6#\"\"#\"$(>\" \"\")-F&6#\"\"$\"#(*F*)-F&6#\"\"&\"#\\F*)-F&6#\"\"(\"#KF*)-F&6#\"#6\"# >F*)-F&6#\"#8\"#;F*)-F&6#\"#\"#5F*)-F&6#\"#B\"\")F*)-F&6# \"#H\"\"'F*)-F&6#\"#JFUF*)-F&6#\"#PF3F*)-F&6#\"#T\"\"%F*)-F&6#\"#VF\\o F*)-F&6#\"#ZF\\oF*)-F&6#\"#`F.F*)-F&6#\"#fF.F*)-F&6#\"#hF.F*)-F&6#\"#n F(F*)-F&6#\"#rF(F*)-F&6#\"#tF(F*)-F&6#\"#zF(F*)-F&6#\"#$)F(F*)-F&6#\"# *)F(F*)-F&6#F/F(F*-F&6#\"$,\"F*-F&6#\"$.\"F*-F&6#\"$2\"F*-F&6#\"$4\"F* -F&6#\"$8\"F*-F&6#\"$F\"F*-F&6#\"$J\"F*-F&6#\"$P\"F*-F&6#\"$R\"F*-F&6# \"$\\\"F*-F&6#\"$^\"F*-F&6#\"$d\"F*-F&6#\"$j\"F*-F&6#\"$n\"F*-F&6#\"$t \"F*-F&6#\"$z\"F*-F&6#\"$\"=F*-F&6#\"$\">F*-F&6#\"$$>F*-F&6#F)F*-F&6# \"$*>F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "200!/10^49 mod 1 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 147 "Les seuls facteurs qui peuvent mettre des z\351ros \+ \340 200! sont 2 et 5. Or 5 est \340 la puissance 49, donc une d\351co mposition de 200! peut aussi s'\351crire " }{XPPEDIT 18 0 "``(10)^49*` `(2)^148*``(3)^97*``(7)^32;" "6#**-%!G6#\"#5\"#\\-F%6#\"\"#\"$[\"-F%6# \"\"$\"#(*-F%6#\"\"(\"#K" }{TEXT -1 100 " \267 \267 \267, de sorte que 200! se termine par 49 z\351ros. On peut v\351rifier le r\351sultat g r\342ce \340 la commande " }{TEXT 282 3 "mod" }{TEXT -1 2 " :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "200!/10^48 mod 10;\n200!/10^ 49 mod 10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Ce dern ier r\351sultat assure encore que 200! / " }{XPPEDIT 18 0 "``(10)^48; " "6#*$-%!G6#\"#5\"#[" }{TEXT -1 35 " est divisible par 10, mais 200! \+ / " }{XPPEDIT 18 0 "``(10)^49;" "6#*$-%!G6#\"#5\"#\\" }{TEXT -1 55 " n e l'est plus, donc 200! se termine bien par 49 z\351ros." }{MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 288 10 "Exercice 6" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 285 1 "a" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 77 "P||1 := x -> x^4-1;\nP||2 := x -> x^3+x^2-5*x-5;\nP ||1(x)/P||2(x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1 Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"%\"\"\"F1F1!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2Gf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,**$)9$\"\"$\"\"\"F1*$)F/\"\"#F1F1*&\"\"&F1F/F1!\"\"F6F7F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"xG\"\"%\"\"\"F)F)!\"\"F) ,**$)F'\"\"$F)F)*$)F'\"\"#F)F)*&\"\"&F)F'F)F*F3F*F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,**$)%\"xG\"\"$\"\"\"F)*$)F'\"\"#F)!\"\"F'F)F)F-F), &F*F)\"\"&F-F-" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " } {TEXT 286 1 "b" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "poly := (x ,y,z) -> (x^2-a)*(x*y+b)*(x^4-y^4)*(3*x+z)*(z-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%polyGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF**, ,&*$)9$\"\"#\"\"\"F4%\"aG!\"\"F4,&*&F2F49%F4F4%\"bGF4F4,&*$)F2\"\"%F4F 4*$)F9F>F4F6F4,&F2\"\"$9&F4F4,&FCF4F2F6F4F*F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "expand(poly(x,y,z));\nsimplify(poly(x,y,z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,R*()%\"xG\"\"(\"\"\"%\"yGF()%\"zG\" \"#F(F(*()F&\"\"$F()F)\"\"&F(F*F(!\"\"*()F&\"\"'F(%\"bGF(F*F(F(**F/F() F&\"\"%F(F6F()F)F9F(F(**F/F(%\"aGF(F%F(F)F(F(**F/F(F " 0 "" {MPLTEXT 1 0 32 "collect(poly(x,y,z), x, fact or);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*&)%\"xG\"\"*\"\"\"%\"yGF(! \"$*&,&%\"bGF**(\"\"#F(F)F(%\"zGF(F(F()F&\"\")F(F(*&,(*&%\"aGF(F)F(\" \"$*(F/F(F-F(F0F(F(*&F)F()F0F/F(F(F()F&\"\"(F(F(*&,(*&F6F(F-F(F7**F/F( F6F(F)F(F0F(!\"\"*&F-F(F:F(F(F()F&\"\"'F(F(*&,(*$)F)\"\"&F(F7**F/F(F6F (F-F(F0F(FA*(F6F(F)F(F:F(FAF()F&FIF(F(*&,(*&F-F()F)\"\"%F(F7*(F/F(FHF( F0F(FA*(F6F(F-F(F:F(FAF()F&FQF(F(*(FPF(F4F()F&F7F(FA*(FPF(F>F()F&F/F(F A*,F6F(FPF(F0F(,&*&F)F(F0F(F(*&F/F(F-F(F(F(F&F(F(**F6F(F-F(FPF(F:F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect(poly(x,y,z), y, factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**,,&*$)%\"xG\"\"#\"\"\" F*%\"aG!\"\"F*F(F*,&F(\"\"$%\"zGF*F*,&F/F,F(F*F*)%\"yG\"\"&F*F**,F%F*% \"bGF*F-F*F0F*)F2\"\"%F*F**,F%F*)F(F3F*F-F*F0F*F2F*F,*,F%F*F5F*)F(F7F* F-F*F0F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect(poly( x,y,z), z, factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*.,&*$)%\"xG \"\"#\"\"\"F*%\"aG!\"\"F*,&*&F(F*%\"yGF*F*%\"bGF*F*,&F(F*F/F,F*,&F(F*F /F*F*,&F&F**$)F/F)F*F*F*)%\"zGF)F*F**2F)F*F%F*F-F*F1F*F2F*F3F*F(F*F7F* F**0\"\"$F*F%F*F-F*F1F*F2F*F3F*F'F*F," }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 287 1 "c" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "coeff(poly(x,y,z), x^5);\ncoeff(poly(x,y,z), y^3);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"yG\"\"&\"\"\"\"\"$*(%\"aGF(% \"bGF(%\"zGF(F(*&,&*&F+F(F,F(!\"$*(F+F(F&F(F-F(!\"\"F(F-F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 290 10 "Exercice 7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eqn := exp(x)-3*x=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&-%$expG6#%\"xG\"\"\"*&\"\" $F+F*F+!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve( eqn, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$-%)LambertWG6##!\"\"\"\" $F(,$-F%6$F(F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Dans ce cas, \+ la fonction " }{TEXT 289 6 "fsolve" }{TEXT -1 99 " ne donne qu'une sol ution, pr\351f\351rez d\351terminer la solution g\351n\351rale, et la \+ convertir en flottants :" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+mGh!> '!#5$\"+_X87:!\"*" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 295 10 "Exercice \+ 8" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 291 1 "a" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "solve(\{ x + 2*y + 3*z + 4*t + 10 = 41, # 1\350re \351quation\n 8*x + 4*z + 3*t + 4 = 11, \+ # 2\350me \351quation\n x + y + z + t + 2 = 9, # 3 \350me \351quation\n 3*y + 4*z - 8*t + 4 = 125 # 4\350 me \351quation\n \}, \{x,y,z,t\}); # vari ables" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"tG#!$P%\"#$)/%\"yG#!#fF (/%\"zG#\"%\"o\"F(/%\"xG#!$/'F(" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 292 1 "b" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 " Ici, on a affaire \340 un syst\350me de plus d'inconnues que d'\351qua tion, il y aura donc 3 variables exprim\351es en fonction de la derni \350re qui peut \352tre n'importe quoi :" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "solve(\{ x + 2*y + 3*z + 4*t + 10 = 41 , # 1\350re \351quation\n 8*x + 4*z + 3*t + 4 = 11, \+ # 2\350me \351quation\n x + y + z + t + 2 = 9 # 3\350me \351quation\n \}, \{x,y,z,t\}); \+ # variables" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"zG,&#\"$V\"\"#7\" \"\"*&#\"#>F)F*%\"tGF*!\"\"/%\"yG,&#F*\"\"'F**&F3F*F.F*F*/%\"xG,&#!#hF )F**&#\"\"&F)F*F.F*F*/F.F." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Qu estion " }{TEXT 293 1 "c" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "solve(\{ x + y + 2*z = 2, # 1\350re \351quation\n 2*x + \+ 3*y + z = 4, # 2\350me \351quation\n x - y + 5*z = 7 \+ # 3\350me \351quation\n \}, \{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yG!\"&/%\"zG#F&\"\"$/%\"xG#\"#JF*" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Question " }{TEXT 294 1 "d" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "solve(\{ x^2 + y^2 = 3, # 1 \350re \351quation\n x^2 + 2*y^2 = 3 # 2\350me \351quation\n \}, \{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG-%'Root OfG6#,&*$)%#_ZG\"\"#\"\"\"F.\"\"$!\"\"/%\"yG\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "allvalues(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"yG\"\"!/%\"xG*$-%%sqrtG6#\"\"$\"\"\"<$F$/F(,$F)!\"\"" }}}} }{SECT 1 {PARA 3 "" 0 "" {TEXT 298 10 "Exercice 9" }}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 9 "Question " }{TEXT 296 1 "a" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(tan(x)-x=0, x=0..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+e%4M\\%!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(tan( x)-x, x=0..10, y=-5..5, discont=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6'7gn7$$\"\"!F)F(7$$\"3^Gh'Q'*))Q U$!#>$\"3;mH?\\6dQ8!#A7$$\"37S7*QW'*HS'F-$\"3mGz[rkyk()F07$$\"3YQVI!Q9 Lv*F-$\"38()fCkk\\/J!#@7$$\"3hwq()RSe78!#=$\"3KZ39wBQ!f(F;7$$\"3?bt(*Q PB[;F?$\"3K]PWd6(*3:!#?7$$\"3'f^&>wRUf>F?$\"3Edpl*p^na#FG7$$\"3_=H*>WV ;G#F?$\"3a*yWYobN/%FG7$$\"3y,.YuK)[h#F?$\"3`fj;y#*\\FhFG7$$\"3Y\"oeDSa q%HF?$\"3LnD&)\\00R))FG7$$\"3[#\\HoBL()G$F?$\"3yJQqDnJR7F-7$$\"3))))[. 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