What is the number of common roots of equation-I and equation-II?

Consider equation-I: z^{3}+2z^{2}+2z+1=0 and equation-II: z^{1985}+z^{100}+1=0.

  1. A. 0
  2. B. 1
  3. C. 2 ✓
  4. D. 3

Correct Answer: C. 2

Explanation

The roots of Equation-I are -1, \omega, \omega^2. Substituting z=-1 in Equation-II gives (-1)^{1985} + (-1)^{100} + 1 = -1 + 1 + 1 = 1 \neq 0. As proven previously, z=\omega is a root, and similarly z=\omega^2 yields (\omega^2)^{1985} + (\omega^2)^{100} + 1 = \omega^{3970} + \omega^{200} + 1 = \omega + \omega^2 + 1 = 0. So there are 2 common roots: \omega and \omega^2.

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