TD 2: polynômes

P n ∈N
P3+P2, P 0P, P +XP 0, P −XP 0P+ (P00)2.
A B
A=X2, B =X+ 1 A=X3+X2+X−1, B =X2+iX + 1
A=X+ 1, B =X2+ 1 A= 2X5+ 4X4−1, B = 2X3+ 2X+ 1,
A=X4+ 3X3−6X2+ 4, B =X2−2.
a
P=X5+X4+aX2−1Q=X3+X+ 1
P∈C[X]a b
P(X−a)(X−b)P(a)P(b)
P= (X+1)5−X5−1X2+X+1
X2+X+ 1
a, b, c, d a 6= 0 x1x2x3
P=aX3+bX2+cX +d x1+x2+x3x1x2x3
P
P X8+ 5X4+ 4
P
P P
R
P=X4+ (3i−2)X3−(1 + 6i)X2+ (4 + 3i)X−2
4P
1 2 P
P

P=X6−X5−4X4−2X3−11X2−X−6
i P
P
X8−1C[X]R[X]
X6+1 C[X]R[X]
X7+√3X5−7
P∈C[X]P
P0
P, Q R
P4−XQ4=XR2.
F
S P (U, V )
UV =P U +V=S.
(X−U)(X−V)U V
p q
z
z3+pz +q= 0
D=4
27 p3+q2d D
z u v
u+v=z, 3uv +p= 0.
U=u3V=v3
P Q U V
z, u, v
z3+pz +q= 0
u+v=z
3uv +p= 0.
j=e4iπ
3
X3+pX +q
p q ∆ = −4p3+ 27q2=−27D
∆>0
∆ = 0
∆<0