Université Montpellier II Fac des Sciences Département de

3:
A B
λ
det(−A) = −det(A)
det(A+B) = det(A) + det(B)
Adet(A)
det(tA) = det(A)
det(AB) = det(BA)
1032
−1 3 0 1
0 4 −2 3
2220
= 90 et
1 0 0 3
1−100
7 5 1 9
2 0 0 1
= 5.
X50 X2−3X+ 2 X2+ 2X+ 1
P∧Q(U, V )∈R[X]2UP +V Q =P∧Q
P=X4+X3−2X+ 1 Q=X2+X+ 1
P=X4−10X2+ 1 Q=X4−4X3+ 6X2−4X+ 1
m, n ∈N∗
r m n (Xm−1) ∧(Xn−1) =
(Xn−1) ∧(Xr−1)
(Xm−1) ∧(Xn−1) = Xm∧n−1
P∈Q[X]X(X−1)P0+P2−(2X+1)P+2X= 0
λ X4−2X3+
λX2+ 2X−1
1 + X+XnC[X]
X8+X4+ 1 R
a, b, c X3−X+ 1 a3+b3+c3
P P (X)−X P (P(X)) −P(X)

3:
(λ, µ)X4+λX3+µX2+ 12X+ 4
R[X]
C[X]R[X] (X2−X+ 1)2+ 1
E={P∈R[X]| ∃Q, R ∈R[X] tels que P=Q2+R2}
E
E P ∀x∈RP(x)≥0