Question

Daca z1 si z2 sunt rădăcinile complexe ale ecuatiei z^2-z+1=0, atunci z1^2015+z2^2015=?

Answer 1

daca inmultesti ecuatia cu z+1 o sa iti dea : z^3+1=0 deci z^3=-1
z1^2015+z2^2015=z1^2*z1^2013+z2^2*z2^2013=-z1^2-z2^2=-(z1+z2)+2
dar z1+z2=1 deci raspunsul ese : 2-1=1

Answer 2

$\displaystyle z^2-z+1=0.~Observam~ca~z=0~nu~este~solutie~a~ecuatiei, \\ \\ deci~putem~inmulti~ecuatia~cu~z,~rezultand:~z^3=z^2-z.~(*) \\ \\ z^2-z+1=0 \Rightarrow z^2-z=-1.~(**) \\ \\ Din~(*)~si~(**)~rezulta~z^3=-1. \\ \\ z_1^{2015}= \left(z_1^3 \right)^{671} \cdot z_1^2=-z_1^2. \\ \\ z_2^{2015}= \left(z_2^3 \right)^{671} \cdot z_2^2=-z_2^2.$

$\displaystyle Deci~z_1^{2015}+z_2^{2015}=- \left(z_1^2+z_2^2 \right)=- \left( \left(z_1+z_2 \right)^2-2z_1z_2\right)=\\ \\ =-(z_1+z_2)^2+2z_1z_2=-1^2+2 \cdot 1=1. \\ \\ (Am~folosit~relatiile~lui~Viete.)$