Question

Stiind ca z1 si z2 sunt radacinile ecuatiei z^2 - z + 1 = 0 calculati:
(1+z1^2)^152 + (1+z2^2)^152

ajutor!!!

Answer 1

$z^2-z+1 = 0 \Big|\cdot(z+1)\\ \\(z+1)(z^2-z+1) = 0\\ \\z^3+1 = 0\\ \\ z^2+1 = z\\ \\ \Rightarrow (z^2+1)^3+1 = 0\Rightarrow (1+z^2)^3 = -1\Big|\verb ^ 50\\ \\\Rightarrow (1+z^2)^{150} = -1 \Big|\cdot (1+z^2) \Rightarrow (1+z^2)^{152} = -1-z^2 =-z\\ \\\\\Rightarrow (1+z_1^2)^{152} + (1+z_2^2)^{152} =\\ \\ =-z_1-z_2 = -(z_1+z_2) =-\dfrac{-1}{1} =\boxed{1}$