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Form 3 Further Factorisation Practice

Form 3 · 20 practice questions with instant marking and worked solutions.

Further Factorisation is an important part of the Form 3 mathematics curriculum. This topic develops common factors, identities and checking factors by expansion. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Further Factorisation has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The twenty original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Further Factorisation; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample.

Effective revision is more than recording a score. After completing the Further Factorisation set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Further Factorisation should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

Start practising

  1. 1 Basic
    Write $x^2-11x+10$ as $(x-1)(x-k)$ and find k. Choose the correct answer.
  2. 2 Basic
    Write $x^2-12x+11$ as $(x-1)(x-k)$ and find k. Give the numerical value only.

  3. 3 Basic
    Write $x^2-13x+12$ as $(x-1)(x-k)$ and find k. Show the essential working and give the numerical value.

  4. 4 Basic
    $x^2-10x+9=(x-1)(x-10)$. Is this true?
  5. 5 Standard
    Write $x^2-13x+22$ as $(x-2)(x-k)$ and find k. Choose the correct answer.
  6. 6 Standard
    Write $x^2-14x+24$ as $(x-2)(x-k)$ and find k. Give the numerical value only.

  7. 7 Standard
    Write $x^2-15x+26$ as $(x-2)(x-k)$ and find k. Show the essential working and give the numerical value.

  8. 8 Standard
    $x^2-12x+20=(x-2)(x-10)$. Is this true?
  9. 9 Advanced
    Write $x^2-15x+36$ as $(x-3)(x-k)$ and find k. Choose the correct answer.
  10. 10 Advanced
    Write $x^2-16x+39$ as $(x-3)(x-k)$ and find k. Give the numerical value only.

  11. 11 Advanced
    Write $x^2-17x+42$ as $(x-3)(x-k)$ and find k. Show the essential working and give the numerical value.

  12. 12 Advanced
    $x^2-14x+33=(x-3)(x-12)$. Is this true?
  13. 13 Challenging
    Write $x^2-17x+52$ as $(x-4)(x-k)$ and find k. Choose the correct answer.
  14. 14 Challenging
    Write $x^2-18x+56$ as $(x-4)(x-k)$ and find k. Give the numerical value only.

  15. 15 Challenging
    Write $x^2-19x+60$ as $(x-4)(x-k)$ and find k. Show the essential working and give the numerical value.

  16. 16 Challenging
    $x^2-16x+48=(x-4)(x-12)$. Is this true?
  17. 17 Elite
    Write $x^2-19x+70$ as $(x-5)(x-k)$ and find k. Choose the correct answer.
  18. 18 Elite
    Write $x^2-20x+75$ as $(x-5)(x-k)$ and find k. Give the numerical value only.

  19. 19 Elite
    Write $x^2-21x+80$ as $(x-5)(x-k)$ and find k. Show the essential working and give the numerical value.

  20. 20 Elite
    $x^2-18x+65=(x-5)(x-14)$. Is this true?

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Further Factorisation

This topic needs particular care with common factors, identities and checking factors by expansion. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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