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Manipulating Rational Expressions Revision Notes | GCSE AQA Higher Further Maths

To simplify an algebraic fraction, factorise the numerator and denominator fully, then cancel any factor that appears in both. Notes and 12 questions answered.

Updated 14 Mar 2026 · Revise Right Team

Manipulating Rational Expressions is part of Manipulation of rational expressions in GCSE AQA Higher Further Maths.

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GCSE AQA Higher

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What this lesson covers

To simplify an algebraic fraction, factorise the numerator and denominator fully, then cancel any factor that appears in both. Notes and 12 questions answered.

Simplify algebraic fractions by factorising numerators and denominators
Add and subtract algebraic fractions with numeric, linear, and quadratic denominators
Multiply and divide algebraic fractions, including those with quadratic expressions
Identify and cancel common factors correctly before performing operations

Key ideas to keep in view

Use this lesson to stay inside Manipulation of rational expressions while connecting the detail back to the wider Algebra topic.

Rational expression: An algebraic fraction where both the numerator and denominator are polynomials. Used when simplifying, combining, or rearranging expressions involving division by algebraic terms.
Common denominator: A shared multiple of two or more denominators. Required when adding or subtracting algebraic fractions so that the fractions can be combined into a single expression.
Cancelling common factors: Dividing both numerator and denominator by the same factor to simplify a fraction. Only valid when the shared factor appears as a multiplied term, not as part of a sum.
Reciprocal: The result of flipping a fraction: the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$. Used when dividing by a fraction, since dividing by $\frac{a}{b}$ is equivalent to multiplying by $\frac{b}{a}$.

Revision notes

Read the core explanations from Manipulating Rational Expressions before testing yourself from memory.

1. Why Rational Expressions Matter

Algebraic fractions appear throughout GCSE Further Maths — in equation solving, proof, and simplification questions. The rules are exactly the same as for numerical fractions, but the numerators and denominators contain algebra instead of plain numbers.

Think about how you add $\frac{1}{3} + \frac{1}{4}$: you find a common denominator, rewrite each fraction, then combine. Every technique in this lesson is that same process, scaled up to expressions like $\frac{x+1}{x-2} + \frac{3}{x+4}$. Once you see the parallel, the method clicks.

2. Simplifying Algebraic Fractions

Before adding, subtracting, multiplying, or dividing, always check whether the fraction can be simplified first. Factorise the numerator and denominator separately, then cancel any factor that appears in both.

Worked example: Simplifying an Algebraic Fraction with a Quadratic Numerator

Simplify $\frac{x^2 - x - 6}{x^2 - 9}$.

$x^2 - x - 6 = (x-3)(x+2)$ — Find two numbers that multiply to $-6$ and add to $-1$: those are $-3$ and $+2$. [M1]
$x^2 - 9 = (x-3)(x+3)$ — Difference of two squares: $a^2 - b^2 = (a-b)(a+b)$ with $a = x$, $b = 3$. [M1]
$\frac{(x-3)(x+2)}{(x-3)(x+3)}$ — Both numerator and denominator are now fully factorised.
$\frac{\cancel{(x-3)}(x+2)}{\cancel{(x-3)}(x+3)} = \frac{x+2}{x+3}$ — $(x-3)$ appears in both, so it cancels. The result is fully simplified. [A1]

Answer: $\frac{x+2}{x+3}$

3. Multiplying and Dividing Algebraic Fractions

Multiplying algebraic fractions follows the same rule as multiplying numerical fractions: multiply the numerators together and multiply the denominators together. The key habit is to factorise and cancel before multiplying, which keeps the algebra manageable.

Division works by the Keep-Change-Flip (KCF) rule: keep the first fraction, change the division sign to multiplication, and flip the second fraction to its reciprocal. Dividing by a fraction always makes the result larger in magnitude — never smaller — so if your answer looks unexpectedly small, check whether you flipped correctly.

Worked example: Dividing Algebraic Fractions

Simplify $\frac{x^2 - 4}{x+1} \div \frac{x-2}{x^2 + 3x + 2}$.

$\frac{x^2 - 4}{x+1} \times \frac{x^2 + 3x + 2}{x-2}$ — Division by a fraction is equivalent to multiplication by its reciprocal. [M1]
$x^2 - 4 = (x-2)(x+2)$ $x^2 + 3x + 2 = (x+1)(x+2)$ — Difference of two squares for the first; find two numbers multiplying to $2$ and adding to $3$ for the second.
$\frac{(x-2)(x+2)}{x+1} \times \frac{(x+1)(x+2)}{x-2}$ — All four expressions are now factorised and ready for cancellation.
$\frac{\cancel{(x-2)}(x+2)}{\cancel{(x+1)}} \times \frac{\cancel{(x+1)}(x+2)}{\cancel{(x-2)}} = (x+2)^2$ — $(x-2)$ and $(x+1)$ each cancel across numerator and denominator. [M1]
$(x+2)^2$ — Expand only if the question asks for it; leaving in factorised form is acceptable unless instructed otherwise. [A1]

Answer: $(x+2)^2$

4. Adding and Subtracting Algebraic Fractions

Adding or subtracting algebraic fractions requires a common denominator, just as with numerical fractions. When denominators are different, you multiply each fraction by a form of 1 (the other denominator over itself) to create equivalent fractions that share the same denominator.

When denominators are linear expressions, the common denominator is their product. When one denominator is quadratic and the other is a factor of it, you only need to scale the simpler fraction — not both. Spotting this saves significant working.

Worked example: Adding Fractions with Linear Denominators

Write $\frac{3}{x+2} + \frac{5}{x-1}$ as a single fraction in its simplest form.

Common denominator: $(x+2)(x-1)$ — The denominators share no common factor, so the common denominator is their product. [M1]
$\frac{3(x-1)}{(x+2)(x-1)} + \frac{5(x+2)}{(x+2)(x-1)}$ — Multiply the numerator and denominator of each fraction by the other's denominator.
$3(x-1) = 3x - 3$ $5(x+2) = 5x + 10$ — Expand carefully, paying attention to signs.
$\frac{3x - 3 + 5x + 10}{(x+2)(x-1)} = \frac{8x + 7}{(x+2)(x-1)}$ — Collect like terms in the numerator. [M1]
$8x + 7$ does not factorise to share a factor with $(x+2)(x-1)$ — The fraction is already in its simplest form. [A1]

Answer: $\frac{8x+7}{(x+2)(x-1)}$

Subtraction follows the same process, but watch the sign carefully when expanding a subtracted numerator. A bracket with a minus sign in front requires every term inside to change sign — this is one of the most common sources of errors in exam questions.

Worked example: Subtracting Fractions with a Quadratic Denominator

Write $\frac{4}{x^2 - 1} - \frac{1}{x+1}$ as a single fraction in its simplest form.

$x^2 - 1 = (x-1)(x+1)$ — Difference of two squares. Recognising this avoids creating an unnecessarily large common denominator. [M1]
Common denominator: $(x-1)(x+1)$ — Since $(x+1)$ is already a factor of $(x-1)(x+1)$, the common denominator is $(x-1)(x+1)$ — no need to multiply both denominators together.
$\frac{4}{(x-1)(x+1)} - \frac{1 \times (x-1)}{(x+1)(x-1)}$ — Only the second fraction needs scaling; multiply its numerator and denominator by $(x-1)$.
$\frac{4 - (x-1)}{(x-1)(x+1)} = \frac{4 - x + 1}{(x-1)(x+1)} = \frac{5 - x}{(x-1)(x+1)}$ — Distribute the minus sign across $(x-1)$: $-(x-1) = -x+1$. Sign errors here are very common. [M1]
$5 - x$ shares no factor with $(x-1)(x+1)$ — The fraction is fully simplified. [A1]

Answer: $\frac{5-x}{(x-1)(x+1)}$

5. Bringing It Together: A Mixed Operation

Exam questions often combine several skills in one problem. You might need to simplify, then add, or factorise a quadratic denominator before finding a common denominator. The strategy is always the same: factorise everything first, cancel what you can, then perform the operation.

Worked example: Combined Operation: Simplify then Subtract

Write $\frac{x^2 + 5x + 6}{x^2 - 4} - \frac{1}{x-2}$ as a single fraction in its simplest form.

$x^2 + 5x + 6 = (x+2)(x+3)$ $x^2 - 4 = (x+2)(x-2)$ — Factorise numerator and denominator of the first fraction before doing anything else. [M1]
$\frac{(x+2)(x+3)}{(x+2)(x-2)} = \frac{x+3}{x-2}$ — Cancel the common factor $(x+2)$. The expression is now much simpler to work with. [M1]
Both fractions now have denominator $(x-2)$ — After simplification, the denominators match, so no scaling is needed.
$\frac{x+3}{x-2} - \frac{1}{x-2} = \frac{x+3-1}{x-2} = \frac{x+2}{x-2}$ — Combine over the shared denominator and simplify the numerator. [M1]
$x+2$ and $x-2$ share no common factor — The fraction is fully simplified. [A1]

Answer: $\frac{x+2}{x-2}$

6. Exam Technique and Common Mistakes

The most mark-losing errors in rational expression questions are predictable and avoidable. Knowing where the traps are is half the battle — the other half is building the habit of factorising before doing anything else.

Sources and content standards

Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.

AQA Level 2 Certificate in Further Mathematics (8365) specification How we check content Report a correction — include this lesson URL

Continue through the topic

Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.

Introduction to Binomial Expansion Introduction to Factorising Formulae and Expressions: Use and Manipulation Factor Theorem with Rational Roots Algebra topic GCSE AQA Higher pathway

Frequently asked questions

How do you simplify algebraic fractions?

To simplify an algebraic fraction, factorise the numerator and denominator fully, then cancel any factor that appears in both. For example, (x² − x − 6)/(x² − 9) = (x − 3)(x + 2)/((x − 3)(x + 3)) = (x + 2)/(x + 3).

How do you add algebraic fractions?

To add algebraic fractions, find a common denominator, rewrite each fraction over it, then combine the numerators. For 3/(x + 2) + 5/(x − 1), the common denominator is (x + 2)(x − 1), giving (3x − 3 + 5x + 10)/((x + 2)(x − 1)) = (8x + 7)/((x + 2)(x − 1)).

How do you multiply algebraic fractions?

To multiply algebraic fractions, multiply the numerators together and the denominators together, as with numerical fractions. The key habit is to factorise every expression and cancel common factors before multiplying, which keeps the algebra manageable.

How do you divide algebraic fractions?

To divide algebraic fractions, use Keep-Change-Flip: keep the first fraction, change division to multiplication and flip the second fraction, then factorise and cancel. For example, (x² − 4)/(x + 1) ÷ (x − 2)/(x² + 3x + 2) simplifies to (x + 2)².

How do you subtract algebraic fractions?

Subtract algebraic fractions over a common denominator, distributing the minus sign across the whole subtracted numerator. For 4/(x² − 1) − 1/(x + 1), factorise x² − 1 = (x − 1)(x + 1), scale only the second fraction, and get (4 − x + 1)/((x − 1)(x + 1)) = (5 − x)/((x − 1)(x + 1)).

Why should you factorise before multiplying algebraic fractions?

Factorising and cancelling before multiplying keeps the algebra manageable. For (x² − 4)/(x + 1) ÷ (x − 2)/(x² + 3x + 2), flipping and factorising gives (x − 2)(x + 2)/(x + 1) × (x + 1)(x + 2)/(x − 2), and cancelling leaves (x + 2)².

How do you simplify (x² − x − 6)/(x² − 9)?

(x² − x − 6)/(x² − 9) simplifies to (x + 2)/(x + 3). The numerator factorises to (x − 3)(x + 2) and the denominator, a difference of two squares, to (x − 3)(x + 3), so the common factor (x − 3) cancels.

How do you find a common denominator for algebraic fractions?

When linear denominators share no common factor, the common denominator is their product. For 3/(x + 2) + 5/(x − 1), the common denominator is (x + 2)(x − 1), giving (3x − 3 + 5x + 10)/((x + 2)(x − 1)) = (8x + 7)/((x + 2)(x − 1)).

What if one denominator is a factor of the other?

When one denominator is quadratic and the other is a factor of it, only the simpler fraction needs scaling. For 4/(x² − 1) − 1/(x + 1), x² − 1 = (x − 1)(x + 1) is already the common denominator, giving (4 − (x − 1))/((x − 1)(x + 1)) = (5 − x)/((x − 1)(x + 1)).

What is the most common mistake when subtracting algebraic fractions?

The most common mistake when subtracting algebraic fractions is not changing every sign in a subtracted numerator. A bracket with a minus sign in front means every term inside changes sign, so −(x − 1) = −x + 1.

How do you simplify then subtract algebraic fractions?

Algebraic fractions are simplified first by factorising everything and cancelling, then subtracted. For (x² + 5x + 6)/(x² − 4) − 1/(x − 2), the first fraction simplifies to (x + 3)/(x − 2), so the answer is (x + 3 − 1)/(x − 2) = (x + 2)/(x − 2).

Are algebraic fractions the same as normal fractions?

Algebraic fractions follow exactly the same rules as numerical fractions, but the numerators and denominators contain algebra instead of plain numbers. Adding 1/3 + 1/4 uses a common denominator, and the same process applies to expressions like (x + 1)/(x − 2) + 3/(x + 4).

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