Finding every complex root once

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Before you start

De Moivre and complex roots. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

Solve $z^{5}=-1$ .

Part a

Give every root in modulus-argument form.

Part b

Find the angular separation of neighbouring roots, as a coefficient of π.

Part c

A student also includes k = 5. Explain why this gives no new root.

Model solution and review criteria

Part a

All moduli are 1. Arguments are (π + 2kπ)/5, for k = 0,…,4. Other equivalent angle ranges are acceptable.

Equate moduli.
Include the 2kπ argument family.
Give n distinct representatives.

Part b

Consecutive arguments differ by $\frac{2\pi}{5}$ .

Subtract neighbouring arguments.

Part c

Its argument differs from k = 0 by 2π, so it represents the same point. The n roots already exhaust the degree-5 equation.

Explain periodicity and completeness.

Compare your own reasoning. No automatic examination marks are awarded.