Polynomial roots — A reciprocal root sum

Question

The roots of are . 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.Combine the fractions to get .
Simplify the previous line carefully, keeping exact values where possible.The root sum is and product is .
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: For a cubic , the product is ; hence the sign is positive.

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