Question

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+x+1=0$. 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 $x^2+x+1=0$

Give $x^2+x+1=0$
$\therefore x=\frac{1}{2}[-1\pm i\sqrt{3}]=\frac{1}{2}(-1+i\sqrt{3}),\frac{1}{2}(-1-i\sqrt{3})=\omega ,{\omega }^2$
But ${\alpha }^{19}={\omega }^{19}=\omega$ and ${\beta }^7={\omega }^{14}={\omega }^2$.
Hence the equation will be same.