Texte du contrôle de mars 2003.

A={v1, v2, . . . , vn}KE
K=R C
a6=b A(X) = X−a B(X) = X−b C(X) = X2+ 1
P(X) = αX2+βX +γ
u, v w P (X) = uA(X)+vB(X)+wC(X)
P(X)K K =R C k > 1
P0(X)
R(X) = X−P(X)
P0(X)
P(X) = (X−z1)(X−z2)· · · (X−zk)zi∈K
i= 1, . . . , k
P(zi)i= 1, . . . , k P 0(zi)6= 0
R(zi)R0(zi)i= 1, . . . , k R0(X)
R
R(X) = A(X)
B(X)A(X)B(X)
deg A= deg B+ 1
R(X) = Q(X)
S(X)
k= deg Q > deg S zi∈K
i= 1, . . . , k
∀i= 1, . . . , k R(zi) = zi, R0(zi) = 0
P(X) = (X−z1)(X−z2)· · · (X−zk)
P0(X)
P(X)=
k
X
i=1
1
X−zi
i= 1, . . . , k S(zi)6= 0 zi
Q(X)−XS(X)i= 1, . . . , k zi
F(X) = 1
R(X)−X

aii= 1, . . . , k
F(X) =
k
X
i=1
ai
X−zi
ai=−1i= 1, . . . , k
R(X) = X−P(X)
P0(X)
R(X) = 2X3
3X2−1
3X2−1
X3−X