Answer to Question #117223 in Algebra for Samuel

Question

10. Given that w denotes either one of the non-real roots of the equation z
3 = 1, show
that
(a) 1 + w + w
2 = 0,

Answer 1

Suppose that $w$ is one of the non-real roots of the equation $z^3=1$ .

$w^3=1, \ \ w^3-1=0, \ \ (w-1)(1+w+w^2)=0 \ \$

$w \ \bcancel{=}\ 1,$ because 1 is real root of the equation $z^3=1$ . So, $w-1 \ \bcancel{=}\ 0.$

We have that $1+w+w^2=0.$