← Complex numbers

7 Polynomial equations with complex roots

This is the question that closes almost every paper: one complex root of a real cubic is given, and you must find the rest — plus an unknown real coefficient — in one pass.

The method, every time

  • A polynomial with real coefficients has complex roots in conjugate pairs: if a+bi is a root, a−bi is automatically a second root — you never have to find it separately.
  • (z − (a+bi))(z − (a−bi)) expands to z² − 2az + (a²+b²), a real QUADRATIC factor you can write down immediately.
  • Divide the cubic by that quadratic factor (or compare coefficients term by term) to find the unknown constant and the real third root together — do not try to guess the third root first.
  • The remainder theorem, f(a) = remainder, opens almost every paper and is pure real algebra: substitute the given root of x − a into the polynomial and solve for the unknown coefficient.
  • A discriminant condition (b² − 4ac = 0 for equal roots, < 0 for no real roots, > 0 for two real roots) turns "how many real roots" into one inequality or equation to solve for a parameter.
Worked example Given one complex root of a cubic One solution of z³ − 8z² + 6z + d = 0 is z = 5 − i, where d is real. Find d and the other two solutions. The conjugate 5 + i is automatically the second root. (z − (5−i))(z − (5+i)) = (z−5)² − i² = z² − 10z + 26, a real quadratic factor. f(z) = (z + B)(z² − 10z + 26) for some real B (leading and constant coefficients must match): comparing the constant term, 26B = d and comparing the z² coefficient confirms the linear factor is (z + 2). So the third root is z = −2, and comparing constants: 26 × 2 = 52 = d. ✓ d = 52; the three roots are 5 − i, 5 + i, and −2. ✕

Given one complex root of a cubic, find the real unknowns first

A recurring Excellence question gives ONE complex root of a cubic with unknown real coefficients (e.g. "z = 6 − 2i is a solution of 2z³ + dz² + 140z − 200 = 0, find d and the other roots"). The route above is always the same: write the real quadratic factor first, then compare coefficients — do not try to guess the third (real) root before you have that factor. ✕

A remainder-theorem question does not need long division

f(x) = (x − a)·q(x) + remainder, so f(a) = remainder directly — substitute the given root straight into the polynomial and solve the resulting linear equation for the unknown coefficient. Setting up polynomial long division for a single-mark Achievement question wastes time you need later in the paper.

Practise just this