Question

Common roots of the equation $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$ 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 $\omega ,{\omega }^2$

$z^3+2z^2+2z+1=0$ ___ (1)

$z^{1985}+z^{100}+1=0$ ___ (2)

We know that $1+\omega +{\omega }^2=0,{\omega }^3=1$

check ω to be root,

(1) ${\omega }^3+2{\omega }^2+2\omega +1$

=$1+\omega +{\omega }^2+1+\omega +{\omega }^2=0+0=0$

(2) ${\omega }^{1985}+{\omega }^{100}+1{\omega }^2+\omega +1=0$

ω is a common root

check ${\omega }^2$ to be a root

(1) ${\omega }^6+2{\omega }^4+2{\omega }^2+1$

= $({\omega }^3{)}^2+2\omega ({\omega }^3)+2{\omega }^2+1$

= $1+\omega +{\omega }^2+1+\omega +{\omega }^2=0$

(2)= ${\omega }^{3970}+{\omega }^{200}+1$

= $\omega +{\omega }^2+1=0$

${\omega }^2$ is a common root

A is correct