The remainder and factor theorems turn division into substitution. If you can find one root of a cubic you can factor it out and reduce the problem to a quadratic, which you already know how to solve completely.
Syllabus points covered
- Use the factor theorem and the remainder theorem
- Divide a polynomial by a linear or quadratic divisor
- Fully factorise a cubic polynomial given one factor, or found by trial
- Solve cubic equations and sketch the resulting curves
- Use the identity for the sum and difference of two cubes
Common mistakes
- Substituting x = a instead of x = −a when testing the factor (x + a).
- Stopping after finding one root. A cubic has three roots, and the quadratic factor usually gives two more.
- Making sign errors in the long-division layout — the most common single source of lost marks in this topic.
- Forgetting the remainder theorem gives the remainder, not the quotient, when the question asks for the full division.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Polynomial long division
- Polynomial long division — answers
- The remainder theorem
- The remainder theorem — answers
- The factor theorem
- The factor theorem — answers
- Solving cubic equations
- Solving cubic equations — answers
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questionsComing soon
- Mark schemeComing soon
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Additional Mathematics past papers