If $\alpha$ and $\beta$ are non-real roots of $x^3-x^2-x-2=0$, then $\alpha^{2020}+\beta^{2020}+\alpha^{2020} \cdot \beta^{2020}=$

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Aicharya · Accepted Answer

Given equation, $x^3-x^2-x-2=0$ $\Rightarrow \quad(x-2)\left(x^2+x+1\right)=0$ $\therefore \alpha$ and $\beta$ are $\frac{-1 \pm \sqrt{3} i}{2}$ or we can say $\alpha$ and $\beta$ are non-real complex roots of unity. So, let $\alpha=\omega$ and $\beta=\omega^2$, where $\omega^3=1$ and $\begin{aligned} & \omega^2+\omega+1=0. \ & \therefore \alpha^{2020}+\beta^{2020}+\alpha^{2020} \beta^{2020} \ & =\omega^{2020}+\omega^{4040}+\omega^{2020} \omega^{4040} \ & =\left(\omega^3\right)^{673} \omega+\left(\omega^3\right)^{1346} \omega^2+\left(\omega^3\right)^{673} \omega\left(\omega^3\right)^{1346} \omega^2 \ & =\omega+\omega^2+\omega^3=1+\omega+\omega^2=1+\alpha+\beta. \end{aligned}$

If \(\alpha\) and \(\beta\) are non-real roots of \(x^3-x^2-x-2=0\), then…

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