A-level · Complex numbers

Complex roots of polynomial equationsExam-style questions

Solving quadratics with a negative discriminant to get a conjugate pair of complex roots; the theorem that complex roots of a real-coefficient polynomial come in conjugate pairs; using one known complex root to find the others and factorise cubics and quartics; and finding the square roots of a + bi by equating real and imaginary parts.

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15 questions · A Level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
It is given that $2 - i$ is a root of the equation $z^4 - 6z^3 + 10z^2 + 2z - 15 = 0$ , whose coefficients are all real. Find the other three roots of the equation. (6 marks)
Level 5 · 6 marksTry it
2
The equation $z^{3} + az + b = 0$ , where $a$ and $b$ are real, has a root $2 + i$ .
(a) Explain why the third root is $-4$ . (3 marks)
(b) Find the values of $a$ and $b$ . (3 marks)
Level 5 · 6 marksTry it
3
It is given that $3i$ is a root of the equation

$z^{4} - 4z^{3} + 14z^{2} - 36z + 45 = 0$
(a) Write down another root. (1 mark)
(b) Find a real quadratic factor of the quartic, and hence find the other quadratic factor. (3 marks)
(c) Solve the equation completely. (2 marks)
Level 5 · 6 marksTry it