Question

Common roots of the equations $z^3 + 2z^2 + 2z + 1 = 0$ and $z^{1985} + z^{100} + 1 = 0$ are

• A. $\omega, \omega^{2}$
• B. $\omega, \omega^{3}$
• C. $\omega^{2}, \omega^{3}$
• D. None of these

Answer 1

Answer: (A)

The given equation $z^{3}+2 z^{2}+2 z+1=0$ can be rewritten as $(z+1)\left(z^{2}+z+1\right)=0 .$
Its roots are $-1, \omega$ and $\omega^{2}$
Let $f(z)=z^{1985}+z^{100}+1$
Putting $z=-1, \omega$ and $\omega^{2}$ respectively, we get
$f(-1)=(-1)^{1985}+(-1)^{100}+1 \neq 0$
Therefore, $-1$ is not a root of the equation $f(z)=0$
Again, $f(\omega) =\omega^{1985}+\omega^{100}+1$
$=\left(\omega^{3}\right)^{661} \omega^{2}+\left(\omega^{3}\right)^{33} \omega+1$
$=\omega^{2}+\omega+1=0$
Therefore, $\omega$ is a root of the equation $f(z)=0$.
Similarly, $f\left(\omega ^{2}\right)=0$
Hence, $\omega$ and $\omega^{2}$ are the common roots.