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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 = \omeg

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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 $\beta^7$.Since $\omega^3 = 1$,we have $\alpha^{19} = \omega^{19} = (\omega^3)^6 \cdot \omega = 1^6 \cdot \omega = \omega$.Similarly,$\beta^7 = (\omega^2)^7 = \omega^{14} = (\omega^3)^4 \cdot \omega^2 = 1^4 \cdot \omega^2 = \omega^2$.The roots of the new equation are $\omega$ and $\omega^2$.Therefore,the required equation is the same as the original equation: $x^2 + x + 1 = 0$.

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + x + 1 = 0$. The equation whose roots are $\alpha^{19}$ and $\beta^7$ is:

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