Polynômes

K[X]
K[X]
K[X]
K[X]
C[X]R[X]
K R C
R
P:x7→
n
X
k=0
akxk,
n∈Na0, a1, . . . , an
x z
a0, . . . , an
•
P:x7→ x2−2x+ 3 (3,−2,1)
Q:x7→ 5−x+ 4x3(5,−1,0,4) n
n
P(3,−2,1,0,0,0. . .) et Q(5,−1,0,4,0,0. . .)
P:x7→ X
k∈N
akxk,

(ak)k∈N
P:x7→ P
k∈N
akxk
Q:x7→ P
k∈N
bkxk(ak) (bk)
p q ∀k≥p+ 1 ak= 0 k≥q+ 1 bk= 0 P Q
∀x∈R
p
X
k=0
akxk=
q
X
k=0
bkxk
N= max(p, q)
∀x∈R
N
X
k=0
akxk=
N
X
k=0
bkxk∀x∈R
N
X
k=0
(ak−bk)xk= 0.
0
(a0−b0)00+ 0 = 0 a0=b0.
∀x∈R
N
X
k=1
(ak−bk)xk=x N
X
k=1
(ak−bk)xk−1!= 0.
x
∀x∈R∗
N
X
k=1
(ak−bk)xk−1= (a1−b1)+(a2−b2)x+···(aN−bN)xN−1= 0.
x0a1=b1a2=b2. . . aN=bN
(ak) (bk)N
R(N)
Φ : 
R(N)→RR
(an)n∈N7→ x7→ P
k∈N
akxk,
(ak)k∈N
(Φ) Φ
ΦR(N)
•
P:x7→ P
k∈N
akxkQ:x7→ P
k∈N
bkxk
p(an)q(bn)
x
P(x) + Q(x) =
p
X
k=0
akxk+
q
X
k=0
bkxk=
max(p,q)
X
k=0
akxk+
max(p,q)
X
k=0
bkxk=
max(p,q)
X
k=0
(ak+bk)xk.

n n
x
P(x)×Q(x) = p
X
i=0
aixi!·
q
X
j=0
bjxj
=
p
X
i=0
q
X
j=0
aibjxi+j=
p+q
X
k=0
ckxk.
ckxkxkaixi
bk−ixk−ii≤k
∀k∈J0, p +qKck=
k
X
i=0
aibk−i.
•
P:x7→ ax2+bx +c,
a b c a 6= 0 P
∃α1, α2∈CP(α1) = P(α2) = 0,
P
∀x∈RP(x) = a(x−α1)(x−α2).
K[X]
K K
0
(1,0,0,0, . . .) 1 1
(0,1,0,0, . . .)X
n∈Nn1
Xnn X1=X X0= 1
K K[X]

P= (an)n≥0
P=X
k∈N
akXk,
k∈Nakk
Pk∈NakXkPk∈NbkXk
X
k∈N
akXk=X
k∈N
bkXk⇐⇒ ∀k∈Nak=bk.
P= (an)n∈N
p∈N
P= (a0, a1, a2, . . . , ap,0,0,0, . . .)
=a0(1,0,0,0, . . .) + a1(0,1,0,0, . . .) + a2(0,0,1,0,0. . .) + ··· +ap(0,0,...,0,1
|{z}p
,0,0, . . .)
=a0+a1X+a2X2+··· +apXp
=
p
X
k=0
akXk=X
k∈N
akXk.
a= (an)b= (bn)KNλ, µ K
λu +µv
λu +µv = (λun+µvn)n∈N.
P=Pk∈NakXkQ=Pk∈NbkXkK[X]λ, µ ∈KλP +µQ
λP +µQ =X
k∈N
(λak+µbk)Xk.
P=X2+X+ 3 Q=X3−2X+ 4 2Q−P

P=Pk∈NakXnQ=Pk∈NbkXkK[X]
P Q P ×Q P Q (ck)k≥0=Pk≥0ckXk
k∈N
ck=
k
X
i=0
aibk−i.
(ck)P Q
1
P, Q, R K[X]λ, µ ∈K
P Q =QP
P·0K[X]= 0K[X]= 0K[X]·P
P·1 = P= 1 ·P
P(Q+R) = P Q +P R = (Q+R)P
(P Q)R=P(QR)
(λP )(µQ)=(λµ)(P Q)
P, Q, R K[X]λ, µ ∈K
[A]kk∈Nk A
k
[P Q]k=
k
X
i=0
[P]i[Q]k−i=
j=k−i
k
X
j=0
[P]k−j[Q]j=
k
X
i=0
[Q]i[P]k−i= [QP ]k.
0K[X]
[0K[X]·P]k=
1) [P·0K[X]]k=
k
X
i=0
[P]i[0K[X]]k−i=
k
X
i=0
[P]i·0=0.
1(= 1K[X])
i j
δi,j =1i=j
0i6=j