Multiple choice

If the cube roots of unity are 1, $w, w^{2}$ then the roots of the equation $(x-1)^{3}+8=0$ are

  1. $-1,-1+2w,-1 -2w^{2}$

  2. -1 , -1, - 1
  3. $-1,1-2w,1-2w^{2}$
  4. $-1,1+2w,1+2w^{2}$
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C Correct answer
Explanation

(x-1)^3 = -8. x-1 = cube root of -8. The cube roots of -8 are -2, -2w, -2w^2. So x = 1-2, 1-2w, 1-2w^2. The roots are -1, 1-2w, 1-2w^2.

AI explanation

By taking the cube root of both sides of the equation (x-1)^3 = -8, we get x - 1 = -2, -2w, or -2w^2. Adding 1 to both sides of each possibility yields the three roots. The roots are x = -1, 1 - 2w, and 1 - 2w^2.

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