Question

Let $\alpha ,\beta$ are roots of the equation $x^2+x+1=0,$ then 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

Answer: (D)

$x^2+x+1=0$

Since the roots are
$w,w^2$ ...cube roots of unity.
Hence
Let
$\alpha =w$
$\beta =w^2$
Therefore
${\alpha }^{19}$
$=w^{19}$
$=w^{18}.w$
$=w$ since $w^3=1$
and
$w^{14}$
$=w^{12}.w^2$
$=w^2$
Hence
$x^2-(w+w^2)x+w^3=0$
$x^2+x+1=0$