Question

Sum of common roots of the equations

$z^3$ + 2$z^2$ + 2z + 1 = 0 and $z^{100}$ + $z^{32}$ + 1 = 0 is equal to :

• A. 0
• B. -1
• C. 1
• D. None

Answer 1

The correct option is B

-1

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

= (z + 1){$z^2$ + z + 1} = 0 $\Rightarrow$ z = -1, $\omega$, ${\omega }^2$.

Let f(z) = $z^{100}$ + 1

$f\left(\omega \right)$ = ${\omega }^{100}$ + ${\omega }^{64}$ + 1 = $\omega$ + ${\omega }^2$ + 1 = 0

$f\left({\omega }^2\right)$ = ${\omega }^{200}$ + ${\omega }^{64}$ + 1 = ${\omega }^2$ + $\omega$ + 1 = 0

∴ Common roots are $\omega$ and ${\omega }^2$ whose sum is -1.