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2022 HSC Mathematics Extension 1

Question 3:
Polynomial roots

1 markpolynomial roots

Let be a polynomial of degree 5. When is divided by the polynomial , the remainder is .

Which of the following is true about the degree of ?

  1. The degree must be 1.
  2. The degree could be 1.
  3. The degree must be 2.
  4. The degree could be 2.

How to recognise this question

Question type: polynomial roots

  • The command and notation point to polynomial roots.
  • The requested response is a multiple choice.

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

  1. By the division algorithm, with either or .
  2. Here is nonzero of degree 1, so . In particular cannot be 1.
  3. Any integer degree (and if is a non-constant polynomial of non-negative degree, or higher if is disallowed by context) is possible in principle; degree 2 is allowed because .
  4. Thus the degree need not be uniquely 2, but it could be 2. Options A–C are false.
  5. Answer: D. (NESA multiple-choice key: D.)

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark 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

When a cubic is divided by , the remainder is . Which statement is true?

  1. must be 1
  2. could be 1
  3. must be 2
  4. could be 2
See the step-by-step answer guide →

Question 2

A degree-4 polynomial leaves remainder 7 on division by . Which is true?

  1. must be 0
  2. could be 1
  3. must be 1
  4. cannot be 1
See the step-by-step answer guide →

Question 3

Remainder occurs on division by . Which is true?

  1. could be 2
  2. must be 3
  3. could be 3
  4. could be 1
See the step-by-step answer guide →