Question

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

• A. $1$
• B. $2020$
• C. $1+\alpha +\beta$
• D. $-1$

Answer 1

Answer: (C)

Given equation, $x^3-x^2-x-2=0$
$\Rightarrow (x-2)\left(x^2+x+1\right)=0$
$\therefore \alpha$ 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 ${\omega }^2+\omega +1=0$
$\therefore {\alpha }^{2020}+{\beta }^{2020}+{\alpha }^{2020}{\beta }^{2020}$
$={\omega }^{2020}+{\omega }^{4040}+{\omega }^{2020}{\omega }^{4040}$
$={\left({\omega }^3\right)}^{673}\omega +{\left({\omega }^3\right)}^{1346}{\omega }^2+{\left({\omega }^3\right)}^{673}\omega {\left({\omega }^3\right)}^{1346}{\omega }^2$
$=\omega +{\omega }^2+{\omega }^3=1+\omega +{\omega }^2=1+\alpha +\beta$