Multiple choice

If $z_1, z_2, z_3, z_4$ are roots of the equation $z^4=1$ then the value of $\sum { i=1 }^{ 4 }{ { z }{ i }^{ 3 } } $ is?

  1. $0$
  2. $1$
  3. $i$
  4. $1 + i$
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A Correct answer
Explanation

The roots of z^4 = 1 are the fourth roots of unity: 1, -1, i, -i. The sum of their cubes is 1^3 + (-1)^3 + i^3 + (-i)^3 = 1 - 1 - i + i = 0.

AI explanation

The roots of the equation z to the fourth power equals 1 satisfy the polynomial z to the fourth power minus 1 equals 0. By Vieta's formulas, the sum of the roots taken three at a time for this polynomial is 0. Therefore, the value of the sum from i equals 1 to 4 of z sub i cubed is 0.

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