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Variation 2 · answer guide

Polynomial roots — Constant remainder

Question

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

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.Constant remainder has degree or is treated as degree in the inequality sense; any works.
  2. Simplify the previous line carefully, keeping exact values where possible.In particular degree 1 is possible (ordinary linear divisor).
  3. Finish the calculation, then check that the result meets the question’s conditions.It is not forced to be any single value.
  4. Match the result to the choices and state the correct option.Therefore option B is correct.

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

Verification: Constant remainder is compatible with every positive divisor degree.

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

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