Polynomial roots — A transformed root expression

Question

If are roots of , find .

This practice question was inspired by Question 14(e) 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. and .
Simplify the previous line carefully, keeping exact values where possible..
Finish the calculation, then check that the result meets the question’s conditions..
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: The result is positive, as a sum of two real squares should be.

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