Key Idea: This topic finds powers and roots of complex numbers with De Moivre's theorem. It also uses conjugate pairs to finish polynomials. It is on Paper 1 and Paper 2.
Complex roots come in pairs
- Real coefficients: if a + bi is a root, so is a − bi.
- The pair gives the real factor z² − 2az + (a² + b²).
- Divide it out to find the other roots.
Roots — equally spaced on a circle
- Polar form — Write the number as R cisφ.
- Modulus — Every root has modulus R¹/ⁿ.
- Angles — (φ + 2πk)/n for k = 0, 1, …, n − 1.
- Check — The n roots form a regular polygon, 2π/n apart.
Change to polar form before any power or root.
3 − i is a root. Which quadratic? z² − 6z + 10.
(1 − i)⁴? (√2)⁴ cis(−π) = −4.
How many fifth roots does a number have? Exactly 5.
Cube roots of 8? 2, −1 + √3 i and −1 − √3 i.
Does the pair rule work with complex coefficients? No. It needs real coefficients.