Question

The common roots of the equations $x^3+2x^2+2x+1=0$ and $x+x^2+x^3=0$ are ( $\omega$ is non real cube root of unity)

• A. $\omega ,{\omega }^2$
• B. $1$
• C. $\omega +{\omega }^2$
• D. $-1$

Answer 1

The correct option is A $\omega ,{\omega }^2$

The roots of $x+x^2+x^3=0$ are $0,\omega ,{\omega }^2$

Let $f\left(x\right)=x^3+2x^2+2x+1$

$f\left(0\right)=1$

$f\left(\omega \right)=1+2{\omega }^2+2\omega +1=0$

$f\left({\omega }^2\right)=1+2\omega +2{\omega }^2+1=0$

Hence, $\omega$ and ${\omega }^2$ are common roots of both the equations.