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Rational Expressions

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Edexcel IAL Maths Pure 3

Rational Expressions

Master algebraic fractions, simplification, and polynomial division with deep understanding and worked examples.

What You'll Learn

  • What rational expressions are and why they matter
  • How to simplify algebraic fractions by factoring
  • Recognizing improper (top-heavy) fractions
  • Using polynomial division to rewrite improper fractions
  • Dividing by quadratic divisors step-by-step

What Are Rational Expressions?

The Big Idea

A rational expression is an algebraic fraction — a ratio of two polynomials. Just like the rational number 3/4 is a ratio of two integers, a rational expression like x/(x+1) or (x²+2x)/(x-3) is a ratio of two algebraic expressions.

The word "rational" comes from "ratio". If you can write something as a fraction (a ratio of two things), it's rational.

Key Point: A rational expression is simply an algebraic fraction. The top and bottom are polynomials, and you manipulate them using the same rules you use for normal fractions.

Why Do We Study This?

Rational expressions appear everywhere in A-Level maths:

  • Partial fractions decomposition — you need to simplify rational expressions first
  • Integration — rational functions are integrated using techniques that depend on simplification
  • Asymptotes and curve sketching — you need to understand the structure of rational expressions to find asymptotes
  • Real-world problems — rates, concentrations, and many physical laws involve rational functions

Examples of Rational Expressions

x (x² + 1) (2x³ - 5x + 2) ─── , ─────────── , ───────────────── x+1 x(x² - 4) (x-1)(x+2)(x-3) These are ALL rational expressions — fractions where numerator and denominator are polynomials.

Simplifying Rational Expressions

The Three-Step Strategy

To simplify a rational expression, always follow this order:

STEP 1: Factorise both top and bottom
Use the factor theorem, algebraic division, or quadratic factorising to completely break down the numerator and denominator into linear and/or quadratic factors.
STEP 2: Cancel common factors
Cancel any linear factors that appear in both the numerator and denominator. Only linear factors (like (x-1), (x+2)) can be cancelled — you cannot partially cancel quadratic factors.
STEP 3: Check if it's improper (top-heavy)
After simplification, check if the degree of the numerator is greater than or equal to the degree of the denominator. If so, you may need to use polynomial division.

The Factor Theorem — Your Key Tool

To factorise a polynomial, you often need the factor theorem:

Factor Theorem
If f(a) = 0, then (x − a) is a factor of f(x)

In other words: if you plug in a number and get zero, then the corresponding linear factor divides the polynomial evenly. This lets you:

  • Test small integers (±1, ±2, ±3, ...) to find one factor
  • Use algebraic division to find the remaining factors
  • Build up the complete factorisation

Worked Example 1: Simplifying by Factoring

Simplify (x³ − 7x + 6) / (x² + 2x − 3)
Step 1: Factor the numerator

Let f(x) = x³ − 7x + 6

Test x = 1: f(1) = 1 − 7 + 6 = 0 ✓

So (x − 1) is a factor.

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Also in the full note
  • Improper Algebraic Fractions
  • Dividing by Quadratic Divisors
  • What to Memorise
  • Concepts Checklist
  • Practice Question 1
  • What Makes a Fraction "Improper"?
  • Why Simplify Improper Fractions?
  • Polynomial Division — Rewriting Improper Fractions
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More Pure Mathematics 3 topics

Rational Expressions
Exponential Growth & Decay
Exponential & Logarithms
Language of Functions
Strategy for Further Trigonometric Equations
Trigonometric Proof
Definitions of Reciprocal Trigonometric Functions
Compound Angle Formulae
Combinations of Transformations
Reciprocal & Inverse Trigonometric Functions