If ω is a complex cube root of unity, find the value of (2 + 2ω + 5ω2)3.
- 2 + 2ω = 2(1 + ω) = 2(−ω2) = −2ω2, because 1 + ω = −ω2.
- So the bracket is −2ω2 + 5ω2 = 3ω2.
- (3ω2)3 = 27ω6 = 27(ω3)2 = 27 × 1.
Answer: 27
Roots of unity and polynomial theorems
Maths · Quadratic equations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
The cube roots of unity are the three solutions of x3 = 1. One is the real number 1. The other two are complex and are written ω and ω2, because each is the square of the other.
The fourth roots of unity are the four solutions of x4 = 1, namely 1, −1, i and −i.
For polynomials, the remainder theorem gives the remainder on division by x − a without doing the division, and the factor theorem tells you when x − a is a factor. Synthetic division is a quick method that gives the quotient and the remainder using the coefficients only.
Rules to remember
- Cube roots of unity: 1, ω = (−1 + √3 i)/2 and ω2 = (−1 − √3 i)/2.
- 1 + ω + ω2 = 0 (sum of the cube roots) and ω3 = 1 (product of the cube roots is 1).
- For a power of ω, divide the power by 3 and keep the remainder: ω3n = 1, ω3n+1 = ω, ω3n+2 = ω2.
- Useful forms: 1 + ω = −ω2, 1 + ω2 = −ω, ω + ω2 = −1.
- Fourth roots of unity: 1, −1, i, −i. Their sum is 0 and their product is −1.
- Remainder theorem: when f(x) is divided by x − a, the remainder is f(a).
- Factor theorem: x − a is a factor of f(x) if and only if f(a) = 0.
Common mistake
With the divisor x + a students substitute x = a. The divisor x + 1 is x − (−1), so the remainder is f(−1), not f(1). In synthetic division, also write 0 for any missing power of x.
Worked example
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