The sum of the 162th power of the roots of the equation x^3-2x^2+2x-1=0 is Answer: 3

Question

The sum of the 162th power of the roots of the equation $x^3-2x^2+2x-1=0$ is

Answer: 3

Accepted Answer

$x^3-2x^2+2x-1=(x-1)(x^2-x+1)=0.$

So roots are $1,\ -\omega,\ -\omega^2$ (since $x^2-x+1=0$ gives $x=\dfrac{1\pm\sqrt3 i}2=-\omega,-\omega^2$).

$$1^{162}+(-\omega)^{162}+(-\omega^2)^{162}=1+\omega^{162}+\omega^{324}=1+(\omega^3)^{54}+(\omega^3)^{108}=1+1+1=3.$$

Solutions · 1