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 $z^3+2z^2+2z+1=0$ 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
$f(-1)=(-1{)}^{1985}+(-1{)}^{100}+1\ne 0$
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.