For the equation x^2 + x + 1 = 0,if alpha and | vedclass

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(D) The given equation is $x^2 + x + 1 = 0$. The roots of this equation are the complex cube roots of unity,$\omega$ and $\omega^2$. Let $\alpha = \om

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$x^2 + x + 1 = 0$

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(D) The given equation is $x^2 + x + 1 = 0$. The roots of this equation are the complex cube roots of unity,$\omega$ and $\omega^2$. Let $\alpha = \omega$ and $\beta = \omega^2$. We need to find the equation whose roots are $\alpha^{19}$ and $\beta^7$. $\alpha^{19} = \omega^{19} = (\omega^3)^6 \cdot \omega = 1^6 \cdot \omega = \omega$. $\beta^7 = (\omega^2)^7 = \omega^{14} = (\omega^3)^4 \cdot \omega^2 = 1^4 \cdot \omega^2 = \omega^2$. Since the new roots are $\omega$ and $\omega^2$,the required equation is the same as the original equation: $x^2 + x + 1 = 0$.

For the equation $x^2 + x + 1 = 0$,if $\alpha$ and $\beta$ are the roots,then which of the following equations has roots $\alpha^{19}$ and $\beta^{7}$?

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