Multiple choice

The common roots of the equations $x^{3}+2x^{2}+2x+1=0$ and $x+x^{2}+x^{3}=0$ are ( $\omega$ is non real cube root of unity)

  1. $\omega,\omega^{2}$
  2. $1$
  3. $\omega+\omega^{2}$
  4. $-1$
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A Correct answer
Explanation

The first equation x^3 + 2x^2 + 2x + 1 = 0 factors to (x+1)(x^2 + x + 1) = 0. The second equation x + x^2 + x^3 = 0 factors to x(1 + x + x^2) = 0. The common roots are the roots of x^2 + x + 1 = 0, which are the complex cube roots of unity, omega and omega^2.

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