Polynomial roots — Build a polynomial from roots

Question

Find the monic cubic whose roots are , and .

This practice question was inspired by Question 11(f) in the 2025 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: translate the requested symmetric expression into sums and products of roots from the coefficients.Write .
Simplify the previous line carefully, keeping exact values where possible.First .
Finish the calculation, then check that the result meets the question’s conditions.Multiply by to get .
State the final answer clearly in the form the question requested.Final answer:

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Substituting each of gives zero.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.