Les polynômes - Institut de Mathématiques de Toulouse

1
◦
∼
2
P Q
P=X4+ 6X3+ 8X2−6X−9Q=X4+X3−X−1
P=X5+X4+X3−X2−X−1Q=X5−3X4+ 5X3−5X2+ 3X−1
P=X5+X+ 1 Q=X4+X+ 1
R[X]
R[X]
P|Qdeg(P) = deg(Q)λ∈RP=λQ
P Q P =Q
P Q ∈R[X]P|Q Q |P
P∈R[X]
P X −a
P(X−a)(X−b)a6=b
a=b
a X4−X+a X2−aX + 1
a, b X2+ 2 X4+X3+aX2+bX + 2
n>1Xn+X+ 1 (X−1)2
P z ∈C\R
z P
z e z
(X2−2<(z)X+|z|2)e∈R[X]PR[X]
P c2nX2n+
c2(n−1)X2(n−1) +· · · +c2X2+c0r P −r
P(X) = cdXd+cd−1Xd−1+· · ·+c1X+c0d
P Q(X) = c0Xd+c1Xd−1+· · · +cd−1X+cdP
2X4+ 3X3−X2+ 3X+ 2 5X3−2X2−2X+ 5
r6= 0 P1/r Q
P r
P1/r P
P∈C[X]
P(X)=(X−r1)e1× · · · × (X−rn)en, ri∈C, ei∈N∗,∀16i6n,
ri

P ei
P
ripgcd(P, P 0)
ripgcd(P, P 0)
deg(pgcd(P, P 0)) = deg(P)−n
r1, r2, r3∈C
P= (X−r1)(X−r2)2(X−r3)3∈R[X].
D= pgcd(P, P 0)
(X−r1)(X−r2)(X−r3)
P D
(X−ri) 1 6i63
pgcd
X3−3X2+ 4 X3+ 2X2−25X−50 X4+X2+ 1
X5−6X4+ 10X3+ 4X2−24X+ 16 X5−4X4+X3+ 14X2−20X+ 8
8X4−44X3+ 54X2−25X+ 4
C
P(X) = X4−(6 + i)X3+ 6(i+ 2)X2−4(2 + 3i)X+ 8i
R[X]C[X]
X3+ 2X2+ 2X+ 1 X4+X3+X2+X+ 1 2X4−3X3−X2−3X+ 2
X3−3X4+ 1 X6+ 1 X4−3X2−4
X4+ 2X2+ 1 X8+ 2X6+ 3X4+ 2X2+ 1 X9+X6+X3+ 1
R[X]X6−3X5+ 6X4−7X3+ 6X2−3X+ 1
−j
R[X]X4+ 2X3+ 5X2+ 4X+ 4
R[X]X6+ 2X5+ 3X4+ 4X3+ 3X2+ 2X+ 1
P(X)=8X3+ 4X2−2X−1R[X]
R[X]X5−4X4−18X3+ 72X2+ 81X−324
P(X) = 6X5−15X4+ 18X2−6X−3R[X]
(X−1)3(X+ 1)2(X−2)

j=e2iπ
3P(X) = X8+ 2X6+ 3X4+ 2X2+ 1
P P
PC[X]R[X]
P(X) = X4+X3−X2+6 P(1+i)
PR[X]C[X]
P=X4−4X3+ 8X2−8X+ 4 ∈R[X]
pgcd(P, P 0)
P
PC[X]R[X]
P(X)=2X5+ 5X4+ 8X3+ 7X2+ 4X+ 1
pgcd(P, P 0)
PR[X]
P(X) = X4−5X3+ 6X2−5X+ 1 r
r6= 0 r2−5r+ 6 −5
r+1
r2= 0
r+1
r2s=r+1 rQ2
Q P C
X4−5X3+9X2−15X+18 R[X]
C[X] 6X4+X3+ (6i+ 10)X2+ (2 + i)X−(4 + 2i)
C[X]X4−2(i+ 2)X3+ 6(i+ 1)X2−4(2i+ 1)X+ 4i
n∈N∗
C[X]R[X]Xn−1
z∈C(z+ 1)n= (z−1n)
R[X] (X+ 1)n−(X−1)n
P(r) = P0(r)=0 r P
r P 0r P
P∈C[X]P0|P P
P∈R[X] pgcd(P, P 0)=1 P
R[X]
P∈R[X]r∈RP0(r) = P00(r)=0 r P
P∈R[X]P0P
1 + j3
R[X] 2
R[X]
P∈C[X]>2C[X]
P(X) = αX3+βX2+γX +δ∈R[X] 3 r=a
b
pgcd(a, b) = 1 P
a δ b α
P X −1
a−b P (1)

a+b P (−1)
Q(X) = 3X3+ 10X2+ 4X−7
t∈RP(X) = X3−3X+t
t∈RP
t P R[X]
tpgcd(P, P 0)
α∈RP(X) = X4+ (α+ 1)X2+α
i P
X2+ 1 P P X2+ 1
α P R[X]
α P X3−3X2+ 2X
p, q p 6= 0 p
q P (X) = X3+pX +q
X3+pX +q3X2+p
P
P∈R[X]P(X2) =
(X2+ 1)P(X)
1−1P
Q∈R[X]P(X)=(X2−1)Q(X)
deg(P(X2)) deg(P(X))
Q
P∈C[X]XP (X−1) = (X−2)P(X)
0 1 P
r∈CP
r6= 0 r−1P
r6= 1 r+ 1 P
P r 6∈ {0,1}
P
P
0 1 PC
PC[X]cXα(X−1)β
c∈C∗α, β ∈N
P
P∈C[X]P(X2) = P(X)P(X+ 1)
P∈C[X]P(X2) = P(X)P(X−1)

C
(x+y= 11
xy = 30 (x+y= 2 xy = 2 (x+y= 2
xy =−1
C
x+y+z= 4
xy +xz +yz =−20
xyz =−48
x+y+z= 2
xy +xz +yz = 1
xyz = 2
x+y+z=−5
xy +xz +yz = 3
xyz = 9
x, y, z ∈Cσ1, σ2, σ3x, y, z
σ1=x+y+z, σ2=xy +xz +yz, σ3=xyz.
x2+y2+z21
x+1 y+1 z
σ1, σ2, σ3
C
x+y+z= 2
x2+y2+z2= 6
1
x+1 y+1 z=1 2
x+y+z= 0
x2+y2+z2= 1
1
x+1 y+1 z=−1
P(X) = X3+aX2+bX +cC[X]r1, r2, r3
a, b c Q r1r2, r1r3
r2r3
KQ,R C
K[X]
K[X]
1
2K
K
3 4
R[X]R C
Q[X] 3 Q R
R[X]