Question

Find the value of 1 + $\omega$ + ${\omega }^2$ + ${\omega }^3$

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Answer 1

If $\omega$ is the cube root of unity , then ${\omega }^3$ = 1
1 , $\omega$ and ${\omega }^2$ are the roots of the equation $x^3$ -1 = 0
Sum of the roots = 1 + $\omega$ + ${\omega }^2$ = $-\frac{\left(coefficientof{x}^2\right)}{coefficientof{x}^3}$
= $\frac{0}{1}=0$
$\Rightarrow$ 1 + $\omega$ + ${\omega }^2$ = 0
$\Rightarrow$ 1 + $\omega$ + ${\omega }^2$ + ${\omega }^3$ = 0 + 1 = 1