Question

Sum of common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{2015}+z^{100}+1=0$ is

• A. $-1$
• B. $2$
• C. $0$
• D. $1$

Answer 1

The correct option is A $-1$

$z^3+2z^2+2z+1=0$
$z^3+z^2+z^2+z+z+1=0$
$(z^3+z^2+z)+(z^2+z+1)=0$
$z(z^2+z+1)+1(z^2+z+1)=0$
$(z+1)(z^2+z+1)=0$
Thus
$z=-1$ and $z=w,w^2$
Clearly $z=-1$ does not satisfy second equation.
$w^{2015}+w^{100}+1=w^2+w+1=0$
$w^{4030}+w^{200}+1=w+w^2+1=0$
Hence $w,w^2$ are the two common roots of the above given equations.
Now,
$w+w^2$
$=-1$ ....(since $1+w+w^2=0$)
Hence, option 'A' is correct.