The common roots of the equation $z^{3} + 2z^{2} + 2z + 1 = 0, z^{2014} + z^{2015} + 1 = 0$ are
- $\omega, \omega^{2}$
- $1, \omega, \omega^{2}$
- $-1, \omega, \omega^{2}$
- $-\omega, -\omega^{2}$
The roots of z^3 + 2z^2 + 2z + 1 = 0 are z = -1 and the roots of z^2 + z + 1 = 0 (which are omega and omega^2). The roots of z^2014 + z^2015 + 1 = 0 are omega and omega^2 because omega^2 + omega + 1 = 0. Thus, the common roots are omega and omega^2.
In the first equation, z^3 + 2z^2 + 2z + 1 = 0, grouping the terms allows us to factor it as (z + 1)(z^2 + z + 1) = 0. This shows the roots are -1 and the complex cube roots of unity, omega and omega^2. To test these in the second equation, z^2014 + z^2015 + 1 = 0, we note that if z is 1, the equation fails, but since omega^3 = 1, we can rewrite the powers for the complex roots. Since 2014 leaves a remainder of 2 when divided by 3, and 2015 leaves a remainder of 0, substituting omega gives omega^2 + 1 + 1 = 0, confirming it works; the same applies to omega^2. The common roots are omega and omega^2, resulting in omega and omega^2.