A Level Maths: Rational Expressions 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.
- Partial Fractions (OCR)
- Partial Fractions (Edexcel)
- Rational Functions and Polynomial Division (OCR)
- Rational Functions and Polynomial Division (Edexcel)
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:
- Polynomial Division And The Factor Theorem — direct prerequisite for higher-order division
- Quadratic Equations — factorisation underpins all algebraic fraction work
- Indices — needed throughout for manipulating powers
A Level Maths rational expressions is the Year 2 extension of algebraic fraction work. You manipulate, simplify, and divide rational expressions of varying complexity — anything of the form polynomial $/$ polynomial. The techniques here, especially partial fractions, are essential for Further Integration, General Binomial Expansion, and modelling problems.
You simplify algebraic fractions by factorising the numerator and denominator and cancelling common factors. You perform addition, subtraction, multiplication, and division of algebraic fractions using a common denominator approach, just like numerical fractions. Higher-order polynomial division extends Polynomial Division And The Factor Theorem to dividing by quadratic divisors and handling cases that do not divide exactly. The headline technique is partial fractions — decomposing a rational expression into a sum of simpler fractions. You handle three cases: distinct linear factors ($\frac{P(x)}{(ax+b)(cx+d)}$), repeated linear factors ($\frac{P(x)}{(ax+b)^2}$), and combinations. This is the gateway to integrating rational expressions later.
Rational expressions 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 the numerator's degree is greater than or equal to the denominator's, you must do polynomial division FIRST to express as polynomial $+$ proper rational expression before partial fractions; for repeated linear factors $(ax + b)^2$, you need TWO terms in the decomposition: $\frac{A}{ax+b} + \frac{B}{(ax+b)^2}$; do not 'cancel' $x$ from $\frac{x^2}{x^2 + x}$ — this is wrong, factorise first to get $\frac{x}{x+1}$; and check your partial fraction decomposition by recombining and comparing to the original.