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Q.
The common roots of the equations $z^3 - (1 + i)z^2 + (1 + i)z - i = 0$ (where $i = \sqrt{-1} )$ and $z^{2016} + z^{2017} -1 = 0$ are
Complex Numbers and Quadratic Equations
Solution:
Given, $z^3 -(1 + i) z^2 + (1 + i)z - i = 0$
$\rightarrow z^2(z - i) - z(z - i) + (z - i) = 0$
$\Rightarrow (z - i) (z^2 - z + 1) = 0$
$\Rightarrow (z - i) (z + \omega)(z + \omega^2) = 0$
$\Rightarrow z = i, -\omega, - \omega^2$
Now, putting these values in $z^{2016} + z^{2017} - 1$ we have
(i) $(i)^{2016} + (i)^{2016} i - 1 = i^{4\lambda} + ii^{4\lambda} - 1 = i \ne 0$
$\therefore z = i$ is not a common root.
(ii) $(-\omega)^{2016} + (- \omega)^{2017} - 1 = \omega^{3(672)} . \omega - 1$
$ = -\omega \ne 0$
$ \therefore -\omega $ is not a common root.
(iii) $(-\omega^2)^{2016} + (-\omega^2)^{2017} - 1$
$ = \omega^{4032} - \omega^{4034} - 1$
$ = 1 - \omega^2 \ne 0$,
so the given equations have no common roots