Polynômes

K R C
K[X]
Pnn∈NXK
P(X) = anXn+an−1Xn−1+· · · +a1X+a0=
n
k=0
akXk
∀i∈J0, nKai∈Kan6= 0
aii
Pdeg(P)n an6= 0 an
0 deg(0) = −∞
X
•x
•z
•u(θ) = cos(θ)v(x) = ex
K[X]K R[X]C[X]
Kn[X]Kn
Kn[X] = {P∈K[X]/deg(P)6n}

P(X) =
p
k=0
akXkQ(X) =
q
k=0
bkXk
P=Q⇐⇒ p=q
∀i∈J0, pK, ai=bi
•
deg(P+Q)6max(deg(P),deg(Q))
•λ∈Kλ P λP
deg(P) = deg(λP )λ6= 0
P, Q ∈Kn[X]α, β ∈K
αP +βQ ∈Kn[X]
Kn[X]
•
deg(P Q) = deg(P) + deg(Q)
deg(P)6= deg(P) deg(P+Q) = max(deg(P),deg(Q))
P(X) = apXp+· · · +a1X+a0ap6= 0 Q(X) = bqXq+· · · +b1X+b0bq6= 0
P(X)Q(X) = cp+qXp+q+· · · +ckXk+· · · +c1X+c0
∀k∈J0, p +qK, ck=
k
i=0
aibk−i=a0bk+a1bk−1+· · · +akb0
aj= 0 j > p bj= 0 j > q
k k
P(X)Q(X)=0 ⇐⇒ P= 0 Q= 0
A(X)P(X) = A(X)Q(X)
A(X)6= 0 =⇒P=Q

P=
n
k=0
akXk∈K[X]Q∈K[X]
P(Q(X)) P◦Q(X) (P◦Q)(X) = P(Q(X)) =
n
k=0
ak(Q(X))k)
X P Q(X)
P Q K[X] deg(P(Q(X))) = deg(P)×deg(Q)
P P P (X)
P P =
n
k=0
akXkP0
P0(X) =
n
k=1
kakxk−1
n∈N∗XnnXn−1
P, Q ∈K[X]λ, µ ∈K
(P+Q)0=P0+Q0,(λP )0=λP 0,(λP +µQ)0=λP 0+µQ0
P, Q ∈K[X] (P Q)0=P0Q+P Q0
P, Q ∈K[X] (P◦Q)0= (P0◦Q)×Q0
P(k)
k P
P(0) =P, P (1) =P0,∀k>1, P (k)= (P(k−1))0= (P0)(k−1)
k Xn
(Xn)(k)=n(n−1)(n−2) · · · (n−k+ 1)Xn−kk6n
0k > n
deg(P) = n k > n P (k)= 0

K[X]
A B K[X]A B B A
Q∈K[X]
B=AQ
B B
B=λA λ ∈KA B
P∈K[X]λ∈K∗P=1
λ(λP )
Q=µ=1
λλ6= 0
P λP (X)P
P
P
0 1
A∈K[X]B∈K[X] (Q, R)∈K[X]2
A=BQ +R
deg(R)<deg(B)
Q R A B
A=BQ +R
(Q, R)∈K[X]2(Q0, R0)∈K[X]2
A=BQ +R=BQ0+R0
deg(R)<deg(B)
deg(R0)<deg(B)
B(Q−Q0) = R0−R
Q−Q06= 0
deg(R0−R) = deg(B(Q−Q0)) = deg(B) + deg(Q−Q0)>deg(B)
deg(R0−R)6max(deg(R0),deg(R)) <deg(B)
Q−Q0= 0 R−R0= 0
B A A B

P∈K[X]a∈Ka P P(a) = 0
P∈K[X]a∈K
P(a) = 0 ⇐⇒ ∃!Q∈K[X]/ P (X) = (X−a)Q(X)
P X −a
∃!(Q, α)∈K[X]×K/ P (X) = (X−a)Q(X) + α
a P (a) = 0 + α
P(X) = (X−a)Q(X) + P(a)
P X −a P (a)
(X−a)|P P (a) = 0
n
P(a1) = P(a2) = · · · =P(an)=0⇐⇒ ∃!Q∈K[X]/ P (X) = (X−a1)· · · (X−an)Q(X)
P∈K[X]a∈Kak P
(X−a)kP(X−a)k+1 P
∃!Q∈K[X]/ P (X) = (X−a)kQ(X)Q(a)6= 0
a k P ⇐⇒
P(a)=0
P0(a)=0
P(k−1)(a) = 0
P(k)(a)6= 0