The n roots sit equally spaced on a circle, so the rest are one angle away.
The nth roots of 8. Change n and watch them arrange themselves.
Same distance out, equally spaced round.
[r cisθ]n = rn cis(nθ). Raising to a power multiplies the modulus's power and multiplies the angle. Going the other way, an nth root takes the nth root of the modulus and divides the angle, then adds 2π/n repeatedly to find the rest.
The roots always sum to zero when there is no xn−1 term, because they are symmetrically placed about the origin. It is a free check and it catches a wrong angle immediately.
A polynomial with real coefficients cannot have a lone complex root. If 2 + 3i is a root then so is 2 − 3i, because the imaginary parts have to cancel for the coefficients to come out real. On the diagram that is a reflection in the horizontal axis, which is why non-real roots always appear above and below in pairs.
1. One root of a real quadratic is 2 + 3i. Give the imaginary part of the other root.
2. Give the modulus of each cube root of 8.
3. How many degrees apart are the fourth roots of a number?
Finding one root properly, then adding 2π/n for the others rather than starting again each time.
Giving arguments in the range the question asks for.
Stating the conjugate pair explicitly when a real polynomial is involved. It is often a whole mark on its own.
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