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The equation (z + 1)⁷ + z⁷ = 0 has seven roots z1, z2,..., z7. Find (a) the sum of the real parts of all roots and (b) the sum of the imaginary parts of all roots.

1. Re-sum = -7/2, Im-sum = 0
2. Re-sum = 0, Im-sum = 0
3. Re-sum = -7/2, Im-sum = 7/2
4. Re-sum = 7/2, Im-sum = 0

Correct answer: Re-sum = -7/2, Im-sum = 0

Solution

The polynomial 2z⁷ + 7z⁶ +... gives sum of roots = -7/2 by Vieta; the coefficients are real so non-real roots occur in conjugate pairs, making the imaginary-part sum 0.

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