2021 HSC Mathematics Extension 1
Question 11(h):
Polynomial roots
The roots of are and .
What is the value of ?
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
- Common denominator: .
- For : sum of products of roots three at a time equals , and product .
- (Vieta for : , . Here , .)
- Hence the sum of reciprocals is .
- NESA: 2 marks full; 1 for writing the reciprocal sum over a common denominator, or for applying one roots–coefficients relation.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 2 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
Roots of are . Find .
Question 2
For with roots , find .
Question 3
Roots of are . Find .