2023 HSC Mathematics Extension 1
Question 11(c):
Polynomial roots
Consider the polynomial , where and are real numbers.
It is given that is a factor of and that, when is divided by , the remainder is .
Find and .
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
- Factor theorem: .
- ...(1)
- Remainder theorem: .
- ...(2)
- Add (1) and (2): .
- From (1): .
- NESA: 3 marks correct; 2 for two equations in ; 1 for observing or .
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
For , is a factor and . Find and .
Question 2
For , is a factor and remainder on division by is . Find .
Question 3
For , and . Find and .