Question

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

• A. $x^2-x+1=0$
• B. $x^2-x-1=0$
• C. $x^2+x-1=0$
• D. $x^2+x+1=0$

Answer 1

The correct option is D $x^2+x+1=0$

The roots of $x^2+x+1=0$
are $\omega ,{\omega }^2$.
Let $\alpha =\omega ,\beta ={\omega }^2$
Now, ${\alpha }^{19}={\omega }^{19}={\omega }^{18+1}={\omega }^{18}\omega =\omega$
and ${\beta }^7=({\omega }^2{)}^7={\omega }^{12}{\omega }^2={\omega }^2$
Which are the same roots.
$\therefore$ Equation $x^2+x+1=0$