Browse Questions

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

$\begin{array}{1 1}(A)x^2-x+1=0 \\ (B) x^2+x+1=0 \\ (C) x^2-x-1=0 \\(D) x^2-x-1=0 \end{array}$

Toolbox:
• $\omega\:\:and\:\;\omega^2$ are the roots of $1+x+x^2=0$
• $\omega^n=1$, if $n$ is a multiple of 3.
• $\omega^n=\omega$, if $n$ leaves remainder 1 when divided by 3.
• $\omega^n=\omega^2$, if $n$ leaves remainder 2 when divided by 3.
Since it is given that $\alpha,\:\beta$ are roots of $x^2+x+1=0$,
$\alpha=\omega\:\:and\:\:\beta=\omega^2$
$\alpha^{19}=\omega^{19}=\omega$
$\beta^7=(\omega^2)^7=\omega^{14}=\omega^2$
$\therefore$ Eqn., whose roots are $\alpha^{19}\:\:and\:\:\beta^7$ is
$x^2+x+1=0$