Question

The common roots of the equation $Z^3+2z^2+2z+1=0,z^{223}+z^{224}+1=0$ are

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

Answer 1

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

We know that If $z^2+z+1=0$, then $z=\omega$ or ${\omega }^2$. And $w^3=1$..................[1]

Now,

$z^3+2z^2+2z+1=0$

$\Rightarrow (z^3+1)+2z(z+1)=0$

$\Rightarrow (z+1)(z^2-z+1)+2z(z+1)=0$

$\Rightarrow (z+1)(z^2-z+1+2z)=0$

$\Rightarrow (z+1)(z^2+z+1)=0$

$\Rightarrow z=-1,\omega ,{\omega }^2$

Substitute $z=-1,\omega ,{\omega }^2$ in the Eqn. $z^{223}+z^{224}+1=0$

$\omega$ and ${\omega }^2$ satisfy the eqn., But not $-1$

∴ the common roots are $\omega$ and ${\omega }^2$