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Q2=XP 2P, Q ∈K[X]
P◦P=P P ∈K[X]
P∈R[X]P(X2)=(X2+ 1)P(X)
n∈N
(1 + X)(1 + X2)(1 + X4). . . (1 + X2n)
p > 0
1,2,4,8, . . .
(Pn)n∈N∗
P1=X−2∀n∈N∗, Pn+1 =P2
n−2
X2Pn
P∈R[X]
∀x∈R, P (x)≥0
∃(A, B)∈R[X]2, P =A2+B2
P∈C[X]P(0) = 1
∀ε > 0,∃z∈C,|z|< ε |P(z)|<1
P=a0+a1X+··· +anXn∈C[X]
M= sup
|z|=1 |P(z)|
∀k∈ {0, . . . , n},|ak| ≤ M
P(X) = Xn+an−1Xn−1+··· +a1X+a0∈C[X]
ξ P
|ξ| ≤ 1 + max
0≤k≤n−1|ak|
P∈C[X]
P+¯ P
R[X]
P∈R[X]≥2P0
x0P m ≥1
P0

f:R→Rf n
f0n−1
P∈R[X]n= deg P≥2
P0
P
P n + 1 ∈N∗n+ 1
P0n
P2+ 1 C
P∈R[X]
P0
P∈R[X]RP0R
(a, b, c)∈R3X10 +aX9+bX8+cX7+X+ 1
R
P∈R[X]Rα∈R∗
P2+α2C
P∈R[X]Rα P 0+αP
R
P∈R[X]RP
P∈R[X]
P
P n ∈NPR
z
Im(z)
n≤P(z)
P02= 4P P ∈K[X]
(X2+ 1)P00 −6P= 0 P∈K[X]
n Pn∈R[X]
Pn−P0
n=Xn
Pn
P∈K[X]
P(X+ 1) =
+∞
X
n=0
1
n!P(n)(X)

P∈R[X]
∀k∈Z,Zk+1
k
P(t) dt=k+ 1
P∈R[X]a∈R
P(a)>0∀k∈N∗, P (k)(a)≥0
P[a; +∞[
(a, b)∈K2a6=b P ∈K[X]
P(X−a)(X−b)P(a)P(b)
a∈KP∈K[X]
P(X−a)2
P(a)P0(a)
t∈Rn∈N∗
R[X] (Xcos t+ sin t)n
X2+ 1
k, n ∈N∗r k n
XkXn−1Xr
n, m ∈N∗
n m Xn−1Xm−1
pgcdXn−1, Xm−1=Xpgcd(n,m)−1
X−1|X3−2X2+ 3X−2
X−2|X3−3X2+ 3X−2
X+ 1 |X3+ 3X2−2
(λ, µ)∈K2X2+ 2 X4+X3+λX2+µX + 2
P∈K[X]
P(X)−X P (P(X)) −P(X)
P(X)−X P (P(X)) −X
P[n]=P◦. . . ◦P n ≥1
P(X)−X P [n](X)−X
P∈K[X]P(X)−X P (P(X)) −X
A, B ∈K[X]A2|B2A|B
(A, B)∈K[X]2
A B
(U, V )∈(K[X]− {0})2
AU +BV = 0,deg U < deg Bdeg V < deg A

A, B ∈K[X]
A B A +B AB
A, B, C ∈K[X]A B
pgcd(A, BC) = pgcd(A, C)
P(X)=(X−a)(X−b)∈C[X]
P(X)P(X3)
a=b P ∈R[X]a6=b a36=b3
R[X]
P a 6=b a3=b3
R[X]
P=anXn+an−1Xn−1+... +a1X+a0
an6= 0 a06= 0
P r =p/q
p|a0q|an
P= 2X3−X2−13X+ 5
P=X3+ 3X−1
Q[X]
a, b, c K
P=X(X−b)(X−c)
a(a−b)(a−c)+X(X−c)(X−a) b(b−c)(b−a)+X(X−a)(X−b) c(c−a)(c−b)
P=λ(X−a)(X−b)(X−c)+1 λ
n∈Nsin ((2n+ 1)α) sin αcos α
P(X) =
n
X
p=0
(−1)p2n+ 1
2p+ 1Xn−p
xk= cot2βkβk
a= cos(π/9)
Z
a
Ka1, a2, . . . , an∈K
n
X
i=1 Y
j6=i
X−aj
ai−aj
A(X) = Qn
j=1 (X−aj)
n
X
i=1
1
A0(ai)

p q
(Xp−1)(Xq−1) |(X−1)(Xpq −1)
∀n∈NX2|(X+ 1)n−nX −1
∀n∈N∗(X−1)3|nXn+2 −(n+ 2).Xn+1 + (n+ 2)X−n
∀(n, p, q)∈N3,1 + X+X2|X3n+X3p+1 +X3q+2
n∈N
X2+X+ 1 |X2n+Xn+ 1
PR[X]
(X+ 4)P(X) = XP (X+ 1)
P∈C[X]
P(1) = 1, P (2) = 2, P 0(1) = 3, P 0(2) = 4, P 00(1) = 5 P00(2) = 6
P, Q, R ∈C[X]
P+Q+R= 0
p, q, r P, Q, R
P p +q+r
P0Q−Q0P
(P, Q, R)
Pn+Qn=Rn
n≥3
n= 2
P∈Z[X]a, b b > 0√b
√6
(a+√b)n
a+√b P a −√b
a+√b P P =RQ2R
QZ[X]
P∈C[X]
P(X2) + P(X)P(X+ 1) = 0
a P a2
a= 0 a
P∈R[X]\ {0}
P(X2) = P(X)P(X+ 1)
0,1,−j, −j2