Question

If alpha and beta are the complex cube roots of unity show that alpha square +beta square +alpha×beta=0

Answer 1

hope it can helps a little...

Answer 2

Answer:  The proof is done below.

Step-by-step explanation:  Given that α and β are the complex cube roots of unity.

We are to show the following equality :

$\alpha^2+\beta^2+\alpha\beta=0.$

We will be using the following factorization formula :

$a^3-b^3=(a-b)(a^2+ab+b^2).$

Since α and β are the complex cube roots of unity, so let

$\alpha=\omega,~~\beta=\omega^2,~\alpha,~\beta\neq 1,\\\\\\\alpha^3=1,~~\beta^3=1.$

We have

$\alpha^3-\beta^3=1-1\\\\\Rightarrow \alpha^3-\beta^3=0\\\\\Rightarrow (\alpha-\beta)(\alpha^2+\alpha\beta+\beta^2)=0\\\\\Rightarrow \alpha^2+\beta^2+\alpha\beta=0~~~~~~~~~~~~~~~~~{\textup{since }\alpha-\beta\neq0}$

Hence proved.