Question

Se considera x1,x2,x3 radacina polinomului f=X^3+x^2-4X+6 . Calculati X1^3+X2^3+x3^3=-31

Answer 1

f(x1)=x1^3+x1^2-4x1+6=0
f(x2)=x2^3+x2^2-4x2+6=0
f(x3)=2x3^3+x3^2-4x3+6=0
Adunam relatiile:
(*) (x1^3+x2^3+x3^3)+(x1^2+x2^2+x3^2)-4(x1+x2+x3)+18=0
Relatiile  lui Viete:
S1=x1+x2+x3=-1
S2=x1x2+x2x3+x1x3=-4
S3=x1x2x3=-6
Ridicam S1 la patrat:
x1^2+x2^2+x3^2+2(x1x2+x1x3+x3x2)=1, x1^2+x2^2+x3^2=1+8=9
Inlocuim in (*):
x1^3+x2^3+x3^3=-18-4-9=-31