If $z_{1},z_{2},z_{3},z_{4}$ are the roots of the equation $z^{4}+z^{3}+z^{2}+z+1=0$ then $|\sum_{i=1}^{4}z_{i}^{1}|$ cannot be equal to -

The given equation z to the fourth + z cubed + z squared + z + 1 = 0 represents a geometric progression. Multiplying by (z - 1) gives z to the fifth - 1 = 0, meaning the roots are the five fifth roots of unity excluding z = 1. The sum of the roots zi is -1. We want to find the possible values of the absolute value of the sum S = z1 + z2 + z3 + z4. Wait, the expression is the absolute value of the sum of zi to the power of 1, which is simply |z1 + z2 + z3 + z4|. By Vieta's formulas for the original polynomial, the sum of the roots is -1. The absolute value of this sum is |-1| = 1. Since the sum is a constant -1, its absolute value is always 1. It cannot be 0, 2, or 3. It cannot be equal to 0.

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