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← 2022 HSC Extension 1 paper Question 3

Variation 3 · answer guide

Polynomial roots — Quadratic remainder

Question

Remainder occurs on division by . Which is true?

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

This practice question was inspired by Question 3 in the 2022 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Step by step

  1. Set up the solution: translate the requested symmetric expression into sums and products of roots from the coefficients. requires , so degree 3 is possible.
  2. Simplify the previous line carefully, keeping exact values where possible.Degree 2 is not allowed ( if equal).
  3. Finish the calculation, then check that the result meets the question’s conditions.Degree 1 is far too small.
  4. Match the result to the choices and state the correct option.Therefore option C is correct.

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Need when remainder is genuinely quadratic.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.

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