Multiple choice

If $\omega \neq 1$ is the cube root of unity, then roots of the equation $\left ( x-a\omega \right )^{3}-a^{3}=0$ are

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

(x - a*omega)^3 = a^3. Taking cube roots: x - a*omega = a, a*omega, a*omega^2. x = a + a*omega, a*omega + a*omega, a*omega^2 + a*omega. Using 1 + omega + omega^2 = 0, the roots are -a*omega^2, 2a*omega, -a.

AI explanation

We solve the equation by taking the cube root of both sides, yielding x - a*omega = a, a*omega, or a*omega^2. This gives the three equations x = a + a*omega, x = 2a*omega, and x = a*omega + a*omega^2. Using the identity 1 + omega + omega^2 = 0, the first and third roots simplify to x = -a*omega^2 and x = -a, respectively. Thus, the roots of the equation are -a, 2a*omega, and -a*omega^2.

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