Question

Find the common roots of the equations $z^3+2z^2+2z+1=1;z^{1985}+z^{100}+1=0.$

• A. $-\omega$ and $-{\omega }^2$ are the common roots.
• B. $1$ and ${\omega }^2$ are the common roots.
• C. $\omega$ and ${\omega }^2$ are the common roots.
• D. $\omega$ and $1$ are the common roots.

Answer 1

Answer: (C)

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

The first equation may be written as $(z+1)(z^2+z+1)=0;$ its roots are -1,$\omega$ and${\omega }^2.$
The root z=-1 does not satisfy the second equation but $z=w$ and $z={\omega }^2$ satisfy it.
Hence $\omega$ and ${\omega }^2$ are the common roots.
Ans: C