2022 HSC Mathematics Extension 1
Question 13(d):
Polynomial roots
The monic polynomial has degree 3 and roots .
It is given that and .
Find .
How to recognise this question
Question type: polynomial roots
- The command and notation point to polynomial roots.
- The requested response is a worked solution.
How to handle it: translate the requested symmetric expression into sums and products of roots from the coefficients.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Write with , , and let .
- Then , so .
- Use : . Alternatively expand fully to obtain the NESA identity .
- Verify: .
- Hence .
- So .
- NESA: 3 marks; 2 for evaluating the sum of correctly in terms of symmetric sums; 1 for writing via elementary identities or defining .
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
Monic cubic, , . Find .
Question 2
For with roots , prove .
Question 3
If and , find .