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If x = (1 + sqrt(3)*i)/2, find the value of y = x⁴ - x² + 6x - 4.

1. -1 + 2*sqrt(3)*i
2. 2 - 2*sqrt(3)*i
3. 2 + 2*sqrt(3)*i
4. none

Correct answer: -1 + 2*sqrt(3)*i

Solution

Because x satisfies x² - x + 1 = 0, every power of x reduces to a linear form; the polynomial collapses to 4x - 3 = -1 + 2*sqrt(3)*i.

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