A Level Maths: Polynomial Division And The Factor Theorem Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Quadratic Equations — needed to solve the resulting quadratic after polynomial division
- Indices — needed throughout for expanding and simplifying
- GCSE Maths Algebra — comfort with brackets and signs is essential
A Level Maths polynomial division and the factor theorem is the standard toolkit for factorising cubics and higher-order polynomials. When a polynomial does not factorise by inspection, the factor theorem helps you spot one root quickly, then polynomial division reduces the problem to a quadratic you can solve. This unlocks curve sketching, integration, and Rational Expressions later on.
The factor theorem says: if $f(b/a) = 0$, then $(ax – b)$ is a factor of $f(x)$. In practice, you try small integer or rational values for $x$ until $f(x) = 0$, giving you your first factor. Polynomial division then divides the original polynomial by that factor — either by long division or by the box method — leaving a quotient that is one degree lower. For a cubic, this leaves a quadratic which you solve by your usual methods to find the remaining roots. You may also be asked about remainders when a polynomial does not divide exactly, where the remainder theorem gives you the value at a specific point without doing the full division.
Polynomial division and the factor theorem are part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: when trying values for the factor theorem, you only need to try values where the candidate divides the constant term (over the leading coefficient) — there is no need to try random values; placeholder zeros matter in polynomial division, so include $0x^2$ (or whichever degree is missing) when setting up the division; check your factorisation by expanding back out at the end; and remember the factor theorem gives a factor $(ax – b)$, not $(x – b/a)$, unless $a = 1$.