Level 3 Complex Numbers Walkthrough

2025 NCEA Level 3 Complex Numbers Question 3(d)

2025 Paper

← Back to paper

Question

\[ 3z^3 + pz^2 + qz - 8 = 0 \]

One solution is \(z=1+i\). If \(p\) and \(q\) are both real, find the other two solutions of the equation, and the value of both \(p\) and \(q\).

First walkthrough idea

Focus to try first

using conjugate roots, factorising a cubic, and matching coefficients to find \(p\) and \(q\).

Step 1

Use the conjugate-root rule

Real coefficients force non-real roots to come in conjugate pairs.

Show the first step’s working

Real coefficients force non-real roots to come in conjugate pairs.

Key result

\(\,1-i\)

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91577 — Apply the algebra of complex numbers in solving problems.

Method: Factorising a cubic and matching coefficients.

This is Question 3(d) from the 2025 NCEA Level 3 Complex Numbers paper for AS91577 — Apply the algebra of complex numbers in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91577.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise factorising a cubic and matching coefficients. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 3(c) Question 3(e) →