Polynomial roots — Root at 1

Question

Let .

(i) Show .

(ii) Hence factor completely over the reals if possible, starting from a linear–quadratic split.

This practice question was inspired by Question 11(a) in the 2020 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.(i) .
Finish the calculation, then check that the result meets the question’s conditions.(ii) .
State the final answer clearly in the form the question requested.The quadratic factors further, but the required linear–quadratic split is .

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

Verification: Each of satisfies .

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