2020 HSC Mathematics Extension 1
Question 11(a):
Polynomial roots
Let .
(i) Show that .
(ii) Hence, factor the polynomial as , where is a quadratic polynomial.
How to recognise this question
Question type: polynomial roots
- The command and notation point to polynomial roots.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: translate the requested symmetric expression into sums and products of roots from the coefficients.
Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.
Step-by-step answer
- (i) Substitute : .
- NESA 1-mark criterion for (i): provides correct solution (successful substitution showing zero).
- (ii) Because is a root, is a factor. Divide by .
- Polynomial division (or equating coefficients): quotient .
- Check: .
- Hence .
- NESA for (ii): 2 marks for correct factorisation; 1 mark for attempting division by .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Let .
(i) Show that .
(ii) Hence factor as a linear factor times a quadratic.
Question 2
Let .
(i) Show that .
(ii) Hence write with quadratic.
Question 3
Let .
(i) Show .
(ii) Hence factor completely over the reals if possible, starting from a linear–quadratic split.