Question

La subiectul ll exercitiul 2.c)
Cum arăt ca e egal cu 3 ,dupa ce calculezi relatiile lui vieté,e vreo formula ceva? :o

Answer 1

$\displaystyle Scriem~relatiile~lui~Viete~pentru~polinomul~f=X^3-2X^2-2X+1. \\ \\ \sigma_1=x_1+x_2+x_3=2 \\ \\ \sigma_2=x_1x_2+x_2x_3+x_1x_3=-2 \\ \\ \sigma_3=x_1x_2x_3=-1 \\ \\ Avem~(x_2+x_3)(x_3+x_1)(x_1+x_2)= \\ \\ =( \sigma_1-x_1)(\sigma_1-x_2)(\sigma_1-x_3)= \\ \\ =\sigma_1^3- \sigma_1^2(x_1+x_2+x_3)+ \sigma_1(x_1x_2+x_2x_3+x_1x_3)-x_1x_2x_3= \\ \\ =\sigma_1^3-\sigma_1^3+\sigma_1\sigma_2-\sigma_3= \\ \\ =\sigma_1\sigma_2-\sigma_3= \\ \\ =2 \cdot (-2)-(-1)= \\ \\ =-3.$