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(A) The roots of the equation $x^2 + x + 1 = 0$ are the complex cube roots of unity, denoted as $\omega$ and $\omega^2$.Thus, $\alpha = \omega$ and $\

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-$1$

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(A) The roots of the equation $x^2 + x + 1 = 0$ are the complex cube roots of unity, denoted as $\omega$ and $\omega^2$.Thus, $\alpha = \omega$ and $\beta = \omega^2$, where $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.We need to calculate $\alpha^{2026} + \beta^{2026} = \omega^{2026} + (\omega^2)^{2026} = \omega^{2026} + \omega^{4052}$.Dividing the exponents by $3$: $2026 = 3 	imes 675 + 1$, so $\omega^{2026} = \omega^1 = \omega$.$4052 = 3 	imes 1350 + 2$, so $\omega^{4052} = \omega^2$.Therefore, $\alpha^{2026} + \beta^{2026} = \omega + \omega^2$.Since $1 + \omega + \omega^2 = 0$, it follows that $\omega + \omega^2 = -1$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + x + 1 = 0$, then $\alpha^{2026} + \beta^{2026} = $

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