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← 2022 HSC Extension 1 paper Question 3

Variation 1 · answer guide

Polynomial roots — Linear remainder forces deg divisor > 1

Question

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

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. forces , so degree 1 is impossible.
  2. Simplify the previous line carefully, keeping exact values where possible.Degree 2 satisfies and is possible.
  3. Finish the calculation, then check that the result meets the question’s conditions.It need not be exactly 2.
  4. Match the result to the choices and state the correct option.Therefore option D is correct.

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

Verification: Use only the strict degree comparison .

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

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