[PDF] polynômes.pdf Exercice 1 [ 02127 ] [Correction] Soit

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??????? ???P2R[X]???? ???P(X2) = (X2+ 1)P(X)? (1 +X)(1 +X2)(1 +X4):::(1 +X2n)? ?? ??????? ? ?????? ?? ????p >0??????? ?? ????? ?????? ????? ????? ?? P

1=X2??8n2N;Pn+1=P2n2?

???8x2R;P(x)0?

8" >0;9z2C;jzj< "??P(z)<1?

????P=a0+a1X++anXn2C[X]? ?? ????

M= sup

jzj=1 P(z)?

8k2 f0;:::;ng;jakj M?

P(X) =Xn+an1Xn1++a1X+a02C[X]?

??????? ??? ????? ?????? ??P????? jj 1 + max0kn1jakj? ????R[X]? P 0? ?? ??????? ? ????? ??????? ?? ? ???????P2+ 1???? ?????? ??????? ????C? ??????? ???P0??? ??????? ?? (a;b;c)2R3? ??????? ???X10+aX9+bX8+cX7+X+ 1????? ??? ????P2R[X]?????? ? ??????? ??????? ????R? ??????? ??? ???? ????2R??? ??? ??? ????? ?????? ???R?

Im(z)nP(z)?

P nP0n=Xn? ????P2K[X]? ???????

P(X+ 1) =+1X

n=01n!P(n)(X)?

8k2Z;Z

k+1 k

P(t)dt=k+ 1?

????P2R[X]? ?? ??????? ???a2R??????

P(a)>0??8k2N;P(k)(a)0?

??????a2K??P2K[X]? ??P0(a)? ??????t2R??n2N? X ??????n;m2N? pgcdXn1;Xm1=Xpgcd(n;m)1? ???X1jX32X2+ 3X2 ???X2jX33X2+ 3X2 ???X+ 1jX3+ 3X22? ???(;)2K2???? ???X2+ 2??????X4+X3+X2+X+ 2? ????P2K[X]? ??? ??????? P(X)X??????P(P(X))P(X)? ?? ??????? ??? P(X)X??????P(P(X))X? ??????? ???P(X)X??????P[n](X)X ????P2K[X]? ??????? ???P(X)X??????P(P(X))X? ????A;B2K[X]???? ???A2jB2? ??????? ???AjB? ?? ?????? (U;V)2(K[X] f0g)2??? ???

AU+BV= 0;degU ????A;B2K[X]??? ????? pgcd(A;BC) = pgcd(A;C)?

P(X) = (Xa)(Xb)2C[X]

???? ???P(X)??????P(X3)? ???? ? ????R[X]?

P=anXn+an1Xn1+:::+a1X+a0

??????? ???pja0??qjan?

P= 2X3X213X+ 5?

P=X3+ 3X1

P(X) =nX

p=0(1)p2n+ 1 2p+ 1 X np nX i=1Y j6=iXaja iaj? ?? ? ???A(X) =Qn n X i=11A

0(ai)?

(Xp1)(Xq1)j(X1)(Xpq1)? ???8n2N?X2j(X+ 1)nnX1 ???8n2N?(X1)3jnXn+2(n+ 2):Xn+1+ (n+ 2)Xn

8(n;p;q)2N3;1 +X+X2jX3n+X3p+1+X3q+2?

X (X+ 4)P(X) =XP(X+ 1)? ??????? ???P2C[X]???? ??? P(1) = 1;P(2) = 2;P0(1) = 3;P0(2) = 4;P00(1) = 5??P00(2) = 6?

P+Q+R= 0?

P n+Qn=Rn ????n3?????? ?????? ??? ?? ????? ?? (a+pb)n? ??? ?????? ??? a+pb??? ?????? ??P?????apb?????? ?? ???? ?????? a+pb??? ?????? ?????? ??P? ??????? ???P=RQ2????R ??Q????Z[X]?

P(X2) +P(X)P(X+ 1) = 0?

??? ??????? ?? a??? ?????? ??P?????a2????? ????? ??????? ??? ??P2R[X]n f0g??????

P(X2) =P(X)P(X+ 1)

P(X2) =P(X)P(X+ 1)?

P(X2) =P(X1)P(X)?

??? ??????? ????? ?????? ???? ??? P??? ?? ?????? ??

P(X2) =P(X1)P(X)?

???X41???X51???(X2X+ 1)2+ 1? ???X4+X2+ 1???X4+X26???X8+X4+ 1 X

2n2cos(na)Xn+ 1?

x

1;x2??x3?

8< :x+y+z= 1

1=x+ 1=y+ 1=z= 1

xyz=4???8 :x(y+z) = 1 y(z+x) = 1 z(x+y) = 1??? 8< :x+y+z= 2 x

2+y2+z2= 14

x

3+y3+z3= 20

??????x;y;z2C???? ???x+y+z= 0? ??????? 1x 2+1y 2+1z 2=1x +1y +1z 2 ????n2N?? ????Pn=Pn k=0Xk? ?? ??????? ?? ? ??????? Qn k=1sinkn+1? ????P2C[X]??? ??? ??n= degP? 8< :x

2+y2+z2= 0

x

4+y4+z4= 0

x

5+y5+z5= 0?

P(X) =a0Xn+a1Xn1++an2C[X]

???? ????p2N? ?? ???? S p=xp

1++xpn?

8>>>>>>>>>><

>>>>>>>>>:a

0S1+a1= 0

a

0S2+a1S1+ 2a2= 0

a

0Sp+a1Sp1++ap1S1+pap= 0 (0< pn)

a

0Sn+a1Sn1++anS1= 0

a

0Sn+k+a1Sn+k1++anSk= 0 (k >0)?

a

2+b2+c2??a3+b3+c3?????? ????? ??????

a+b+c2R?a2+b2+c22R??a3+b3+c32R? ?????a?b??c???? ????? ?????? f n(x) = cos(narccosx)? ?????? ????]1;1[? ???? ????i2 f0;1;:::;ng?? ???? L i=Q

0jn;j6=i(Xaj)Q

0jn;j6=i(aiaj)?

?????? ?????? ? ??????? j2 f0;1;:::;ng? ?? ?Li(aj) =i;j

8P2Kn[X];P(X) =nX

i=0P(ai)Li(X)? ????(Pn)n0?? ????? ??K[X]?????? ??? P

0= 0;P1= 1??8n2N;Pn+2=XPn+1Pn?

8n2N;P2n+1= 1 +PnPn+2?????? ???????

??????? ? ?????? ? ??????? m2N?? ???? ????n2N?? ? P m+n=PnPm+1Pn1Pm? ??? ??????? ? ??????? m2N?? ???? ????n2N?? ? pgcd(Pm+n;Pn) = pgcd(Pn;Pm)? pgcd(Pn;Pm) =Ppgcd(m;n)? L n(x) = exdndxn(exxn)? ???? Tn1;Tn??Tn+1? P

0= 2;P1=X??8n2N;Pn+2=XPn+1Pn?

P n(z+ 1=z) =zn+ 1=zn? f(x) =1cosx?

8n1;f(n)(x) =Pn(sinx)(cosx)n+1?

??degP1????? ?? ???? ??????P=aX+b?? ?????

PP=P()a(aX+b) +b=aX+b()a2=a

ab=0?

P(X) = (1 +X)(1 +X2)(1 +X4):::(1 +X2n)?

(1X)P(X) = 1X2n+1? ???????P1=X2? ?? ?a1=2?b1= 1??c1= 0? ???????Pn+1=P2n2? ?? ?an+1=a2n2?bn+1= 2anbn??cn+1=b2n+ 2ancn? ?? ?? ??????a2= 2?b2=4??c2= 1???? ????n3?an= 2?bn=4n1? c n= 4n2+ 4n1++ 42n4= 4n24n113 (X)2= (X)2+ 02 X

2+ 2pX+q= (X+p)2+pqp22?

(A2+B2)(C2+D2) = (ACBD)2+ (AD+BC)2

P(z) = 1 +aqzq+zq+1Q(z)

????aq6= 0??Q2C[X]? z n=1n ei()=q?

P(zn) = 1jaqjn

q+ o1n q

P(zn)<1????n????? ??????

P(1) +P(!) ++P(!n) = (n+ 1)a0

nX k=0! k`=( n+ 1??`= 0 [n+ 1]

0??????

?? ?? ??????(n+ 1)ja0j (n+ 1)M????ja0j M?

P(1) +!kP(!) ++!nkP(!n) = (n+ 1)ak

?? ?? ?? ??????jakj M? m= max0kn1jakj? n=an1n1++a1+a0 jjnn1X k=0jakjjjkmn1X k=0jjk jjnmjjn1jj 1mjjnjj 1

P(X) =nY

k=1(Xzk) nY k=1(zzk) = nY k=1(zz k)? ??Imz >0?????

8k2 f1;:::;ng;jzzkj k nY k=1(zzk) n Y k=1(zz k) ?? ?? ?????? ??? ??? ??????? ??P???? ?????? ??????? ?? ????P??? ?????? ???? R[X]? ?? ?? ??? ?? ???? ??ImP? f(a) =f(b)????? ?? ??????c2]a;b[??? ???f0(c) = 0? m

1++mp= degP?

y x

1;:::;xp? ?? ? ????? ?????? ?? ?????

(p1) + (m11) ++ (mp1) = degP1 ??????? ??P0? ??degP0= degP1????P0??? ??????? a

1< b1< a2<< bn1< an

1++p=n?

11;:::;p1?

??????? ?P0? p X i=1 i1 +p1 =n1 = degP0 ?????? a0< a1< ::: < an??? ??????? ??P? a

0< b1< a1< b2< ::: < bn< an

(P2+ 1)0= 2PP0? ?? ??? ??????? ??P?? ???? ??? ??????? ??P2+ 1?? ??? ??????? ??P0???? ??????? ?? ?? ??????? ???? ???? ??????? ??P2+1? ??? ?????P2+1??(P2+1)0 a

1< b1< a2< b2< ::: < bp1< ap?

?? ???b1;b2;:::;bp1???? ??? ??????? ?? ????? ??????? ??P0? ?? ????? ?? ?? ??? ??????? ??P2+2?? ??????? ???? ??????? ?? ??? ??????? ??PP0???? ?????? ??????a0< a1< ::: < am??? ??????? ??????? ??P??0;1;:::;m????? deg(P) =mX k=0 k? m X k=0(k1) = deg(P)(m+ 1) f

0(x) =P0(x) +P(x)ex

a a

0< b1< a1< b2< ::: < bm< am?

????6= 0? ?? ?????? ?????? ??? ?????? ???deg(P0+P) = deg(P)? ??? ??? P P

00;P000;:::

P(X) =+1X

n=0a nXn a p= 0????ap1;ap+1>0?

Q(X) =P(p1)(X) = (p1)!ap1+(p+ 1)!2

ap+1X2+::: ?? ? ??? ??????? ????? ??Q????Q(0)>0?

P= (Xa1):::(Xan)?

???? ????z2C? ?? ? ?????

P(z)=nY

k=1jzakj Im(z)n jzakj=q

Re(z)ak

2+Im(z)2Im(z)?

?????Im(z)n0?? ????z2R? ?????? ??? ??????? ??P???? ?????? ??????? ?? ??

2(degP1) = degP?? ????degP= 2? ?? ???? ????? ??????

P=aX2+bX+c????a6= 0?

P

02= 4P()4a2X2+ 4abX+b2= 4aX2+ 4bX+ 4c()a= 1

c=b2=4 degP(degP1) = 6????degP= 3? P n=anXn+an1Xn1++a1X+a0P a n= 1;an1=nan;an2= (n1)an1;:::;a0= 1:a1?

P=Xn+nXn1+n(n1)Xn2++n! =nX

k=0n!k!Xk?

P(X) =+1X

n=0P (n)(0)n!Xn

P(1) =+1X

n=0P (n)(0)n! P (k)(1) =+1X n=0P (n+k)(0)n!?

P(X+ 1) =+1X

k=0P (k)(1)k!Xk=+1X k=0+1X n=01k!P (n+k)(0)n!Xk

P(X+ 1) =+1X

n=0+1X k=01k!P (n+k)(0)n!Xk=+1X n=01n!P(n)(X)?

8k2Z;Q(k+ 1)Q(k) =k+ 1?

X(X+ 1)???

8k2Z;(Q~Q)(k+ 1) = (Q~Q)(k)

P(X) =X+12

??? ?? ??????? ?? ??????? ?? ? ???? ????x0

P(a+x) =degPX

k=0P (k)(a)k!xkP(a)>0? =P(b)P(a)ba??=bP(a)aP(b)ba?

P=Q(Xa)2+R????degR <2?

?? ???? ??????R=X+?

R=aX+b????a;b2R?

(icost+ sint)n=R(i) =ai +b??(icost+ sint)n= ei(n=2nt)???? a= sinn(=2t)??b= cosn(=2t)? ?? ?k=nq+r????0r < n? ??XkXr=Xr(Xnq1)??Xn1jXnq1? ?? ???? ???? ?????? X nq1 = (Xn1)Q(X) X k= (Xn1)XrQ(X) +Xr????degXrR=Xr1?

X n1???Xm1? a A

0=Xn1?A1=Xm1?? ???? ????k??? ???ak6= 0?Ak6= 0??Ak+1???

??? ?????pgcd(Xn1;Xm1) = pgcd(A0;A1) = pgcd(A1;A2) == pgcd(Ap;Ap+1) =Ap=Xpgcd(m;n)1???Ap+1= 0???????ap+1= 0? ???X32X2+ 3X2 = (X1)(X2X+ 2)? ???X3+ 3X22 = (X+ 1)(X2+ 2X2)?quotesdbs_dbs32.pdfusesText_38

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