Polynômes I L`ensemble K[X] des polynômes à coe cients dans K

K R C
K[X]K
PK K P
n n + 1
a0, a1, a2, . . . , anK
∀x∈K, P (x) = a0+a1x+a2x2+· · · +anxn=
n
X
k=0
P=
n
X
k=0
akXkX
a0, a1, a2, . . . , anP
K[X]K
Q Q = 0
P=
n
X
k=0
akXkQ=
n
X
k=0
bkXk
K[X]
P=
n
X
k=0
akXkQ=
m
X
k=0
bkXkK[X]λ∈K

∀k > n, ak= 0 ∀k > m, bk= 0
K[X]
P+Q=X
P×Q=
n+m
X
k=0
ckXkck=
λ
λ·P=
P=X3+ 3X2+ 1 Q=X2+X
P+Q=P Q =
6·P=
P=
n
X
k=0
akXkK[X]
P6= 0 P(P)
(0) =
P6= 0 P ann= (P)

(2X2−3)3
P Q K[X]
P Q
(P Q) =
(P)6= (Q) (P+Q) =
(P) = (Q) (P+Q)6
n= (P)m= (Q)
P Q R P Q =P R
K[X]
P Q K[X]P Q P
Q Q P P |Q

X3−1
P P
P Q Q 6= 0 (P)
P Q Q P
P Q P |Q Q|P
λ
A B K[X]B6= 0
(Q, R)∈K[X]2
Q R A B

A=X4+ 3X2−X+ 3
B=X2+X+ 1
PK[X] (P)>1
PK[X]λ P λ ∈K∗
X2+ 1 R[X]
X2+ 1 C[X]
K[X]
P=
n
X
k=0
akXkK[X]P
P0P0= 0 P
P0=
3X2−X+ 2