Polynômes à coe cients réels ou complexes 1 Définition

K R C
A
+×
(A, +)
×
×
∀(x, y, z)∈A3x×(y+z) = x×y+y×z
×
∀(x, y, z)∈A3(x+y)×z=x×z+y×z
×
×A
A
(Z,+,×) (C,+,×)
DR
D
D
n(Mn(K),+,×)
KE(L(E),+,◦)
(N,+,×)
R R
C[X]
X
C
C[X]
C⊂C[X]
C[X]X
C[X] +
×
C[X]
0
1
k
XkX×X×. . . ×X X
k
X0= 1 a∈C
k∈NaXk
k
P n
a0anP
n
X
k=0
akXk
P
P
P
P u
u0=a0u1=a1un=an
k≥n uk= 0 P
a0an
P
P∈C[X]P

R[X]
0C
C[X] 0
λ∈Cλ
P∈C[X]\ { 0}(an)n∈N
P
P
deg(P)
adeg(P)6= 0
P
deg(P) + 1 0
deg(P)−1
P
P=
deg(P)
X
n=0
anXn
adeg(P)P
adeg(P)Xdeg(P)P
−∞
k0−∞
n∈N Kn[X]
K
n
0
P
(ak)k∈Nn
k≥n
ak= 0 P
P Q
(ak)k∈N
(bk)k∈Nλ∈C
P+Q
(ak+bk)k∈NP+Q
P
Q
deg (P+Q)≤max (deg(P),deg(Q))
P+Q=
max (deg(P),deg(Q))
X
k=0
(ak+bk)Xk
λP
(λak)k∈N
λP =
deg(P)
X
k=0
λakXk
λdeg (λP ) = deg(P)
λdeg (λP ) = −∞
P Q
(ck)k∈N
∀k∈Nck=
k
X
j=0
ajbk−j
deg (P Q) = deg (P) + deg (Q)
∀n∈N∪ { −∞ } n+
−∞ =−∞ +n=n
d∈Nn∈Nλ1λn
nKP1Pnn
Kd
n
X
k=1
λkPk
K
d
P Q
P Q = 0 ⇐⇒ P= 0 Q= 0
(P, Q)∈K[X]2
n∈N
(P+Q)n=
n
X
k=0 n
kPkQn−k
Pn−Qn= (P−Q)
n−1
X
k=0
PkQn−1−k

(P, Q)∈K[X]2d
P a0adP
P Q P ◦Q
P◦Q=
d
X
k=0
akQk
P=X2+ 2X+ 7
Q=X+ 3 P◦Q
P(Q)P◦Q
P∈K[X]a∈K
d P λ0λn
P a
P a P (a)K
P(a) =
d
X
n=0
λnan
Pe P
e
P:K→K
x7→ P(x)
Pe P
(P, Q)∈K[X]2λ∈K
^
P+Q=e P+e Q
g
P Q =e P×e Q
f
λP =λe P
P∈K[X]a∈K
a P P (a)=0
P D
KD P
P D P
D D|P
QK
P=D×Q
Q
n∈Na∈K
X−a|Xn−an
P∈K[X]a∈Ka
P X −a P
P(a) = 0 ⇐⇒ X−a|P
P∈K[X]n∈N
a1annKa1
anP
Qn
k=1(X−ak)P
P∈K[X]a∈Kr∈N
a r P
P(X−a)r
(X−a)r+1
a0P
P(a)6= 0
a r P r ≥1
P(a)=0
P∈K[X]a∈K
r∈Na r P
P(X−a)rQ Q(a)6= 0
P∈K[X]
Pdeg (P)
n∈NP∈Kn[X]
P n + 1
P
n∈N(P, Q)∈Kn[X]2
P Q n + 1
P=Q
(P, Q)∈K[X]2P
Q
P=Q
(P, Q)∈K[X]2e P=
e
Q P =Q

P∈K[X]d=
deg(P)a0adP
P P 0
P0=
d
X
k=1
akkXk−1
R[X]
∀P∈R[X]g (P0) = e P0
P0−∞ P0= 0
deg (P0) = deg (P)−1
P∈K[X]
n∈Nn P P (n)
P(0) =P
∀n∈NP(n+1) =P(n)0
(P, Q)∈K[X]2
(λ, µ)∈K2
(λP +µQ)0=λP 0+µQ0
(P Q)0=P0Q+P Q0
(P◦Q) = Q0×(P0◦Q)
(P, Q)∈K[X]2n∈N
(P Q)(n)=
n
X
k=0 k
nP(k)Q(n−k)
n∈NP∈Kn[X]
P=
n
X
k=0
P(k)(0)
k!Xk
n∈
NP∈Kn[X]a∈K
P=
n
X
k=0
P(k)(a)
k!(X−a)k
P∈K[X]r∈N
a∈K
a r P
P(r)(a)6= 0 k∈[[0, r−1]] P(k)(a)=0
P∈K[X]r∈N∗a∈K
a r P a
r−1P0
C[X]
C
C
n∈NP
n c
z1zn
P=c
n
Y
k=1
(X−zk)
zkk∈[[1, n]] P
n∈NP∈C[X]
n c
p P z1
zpP n1np
n=
p
X
k=1
nk
P=c
n
Y
k=1
(X−zk)nk
R[X]
P
z∈Cr∈N∗z
r P z
r P
P
P
P
P=c
n
Y
k=1
(X−ak)
m
Y
k=1
(X−zk)(X−zk)
a1anP
z1zm

z1zm
c
P n + 2m= deg (P)
P=c
n
Y
k=1
(X−ak)
m
Y
k=1 X2−2Re(zk) + |zk|2
R
X5+ 1 X4+ 1