Exercices sur les polynômes.

K=R C
P=a0+a1X+ +a2X2+. . . +anXn∈C[X]a0, . . . , an∈Zx=b
c∈Q
b, c x P b|a0c|an
Q
X3−p p ∈Z
X5−X4+X3+X2+X+ 1
2X3−X+ 2
f:R−→ R
∀x∈Rf(x) = sin(x)
∀x∈Nf(x) = sin(x)
∀x∈[0,1] f(x) = sin(x)
P=anXn+. . . +a1X+a0∈C[X]an6= 0
h(P) = max |a0
an
|, . . . , |an−1
an
|
z∈CP|z| ≤ 1 + h(P)
3X5+ 4X2+ 1 X2+ 2X+ 3
X4−X3+X−2X4−2X+ 4
(P, Q)
P=X3−X2−X−2Q=X5−2X4+X2−X−2
P=X4+X3−2X+ 1 Q=X3+X+ 1
P=X3+ 1 Q=X2+X+ 2 U, V ∈K[X]UP +V Q = 1
(U, V )
R[X]P=X4+ 1
R[X]C[X]
X3−2
X5−1

X24 −1X15 −1
X24 −1X18 −1
X12 −1X2+X+ 1
n∈N∗Xn−1 (X−1)n
a∈R(X+ 1)7−X7−a
P∈C[X]
P∈C[X]˜ P∈C[X]P
C
P∈K[X]P0|P a ∈K∗b∈K
n∈N∗P=a(X−b)n