If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then the equation whose roots are $\alpha^{19}$ and $\beta^7$ is

Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then the equation whose roots are $\alpha^{19}$ and $\beta^7$ is, $x^2-x-1=0$, $x^2-x+1=0$, $x^2+x-1=0$, $x^2+x+1=0$

MARKS App · Accepted Answer

Hints: $\alpha$ and $\beta$ are the roots of $x^2+x+1=0$ $\begin{array}{ll} \alpha=\omega & \beta=\omega^2 \ \alpha^{19}=\omega & \beta^7=\omega^2 \ x^2-\left(\alpha^{19}+\beta^7\right) x+\alpha^{19} \beta^7=0 \end{array}$ Thou, $\begin{aligned} & x^2-\left(\omega+\omega^2\right) x+\omega \cdot \omega^2=0 \ & x^2+x+1=0 \end{aligned}$

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