Question

The common roots of the equations $x^{12}-1=0$, $x^4+x^2+1=0$ are

• A. $\pm \omega$
• B. $\pm {\omega }^2$
• C. $\pm \omega ,\pm {\omega }^2$
• D. None of these

Answer 1

Answer: (C)

$x^{12}-1=\left(x^6+1\right)\left(x^6-1\right)$
$=\left(x^6+1\right)\left(x^2-1\right)\left(x^4+x^2+1\right)$
Common roots are given by $x^4+x^2+1=0$
$\therefore x^2=\frac{-1\pm i\sqrt{3}}{2}=\omega ,{\omega }^2$
or ${\omega }^4,{\omega }^2$
$\left(\because {\omega }^3=1\right)$
or $x=\pm {\omega }^2,\pm \omega$