Polynomial roots — Different remainder

Question

For , is a factor and . Find and .

This practice question was inspired by Question 11(c) in the 2023 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.: .
Finish the calculation, then check that the result meets the question’s conditions.: .
State the final answer clearly in the form the question requested.Subtract: , then .

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

Verification: Check and .

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