Polynomial roots — Integer root then factor

Question

Let .

(i) Show that .

(ii) Hence factor as a linear factor times a quadratic.

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) .
Simplify the previous line carefully, keeping exact values where possible.(ii) Divide by : quotient .
Finish the calculation, then check that the result meets the question’s conditions.Check: .
State the final answer clearly in the form the question requested.So .

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

Verification: Expand the product to recover ; both factors are required.

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