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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 ?

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

1 Answer

  • $\omega\:\:and\:\;\omega^2$ 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 $x^2+x+1=0$,
$\therefore$ Eqn., whose roots are $\alpha^{19}\:\:and\:\:\beta^7$ is
answered Jul 24, 2013 by rvidyagovindarajan_1

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