Edexcel IAL P3 Algebraic Fractions: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P3 algebraic methods is about simplifying rational expressions: factorise every numerator and denominator, then cancel common factors, multiply, divide, add or subtract. You can only cancel a factor of the whole numerator and the whole denominator, never a single term.

P3 · Pure Mathematics 3
Calculator allowed
3 worked examples
30 practice questions
Tested in 4 of the last 4 papers

Practise P3 algebraic fractions →

Notes
Worked examples
Mistakes
FAQ

P3 Algebraic Fractions notes

Simplifying by cancelling

Factorise fully, then cancel factors that appear top and bottom: $\frac{x^{2} - 9}{x^{2} + x - 6} = \frac{(x - 3) (x + 3)}{(x + 3) (x - 2)} = \frac{x - 3}{x - 2}$ .

Multiplying and dividing

Multiply numerators and denominators, cancelling first to keep things small. To divide, multiply by the reciprocal of the second fraction: $\frac{2 x}{x + 1} \div \frac{4 x^{2}}{x^{2} - 1} = \frac{x - 1}{2 x}$ .

Adding and subtracting

Use the lowest common denominator, then simplify the numerator: $\frac{1}{x + 1} + \frac{2}{x - 1} = \frac{3 x + 1}{x^{2} - 1}$ . Put brackets round each numerator before you subtract.

Improper fractions and algebraic division

When the numerator's degree is at least the denominator's, divide first: $\frac{x^{2} + 3 x + 5}{x + 1} = x + 2 + \frac{3}{x + 1}$ . This form is also the first step for partial fractions in P4.

P3 Algebraic Fractions worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 5 marks · hard

Express $\frac{x + 2}{x^{2} - 5 x + 6} - \frac{x - 1}{x^{2} - 4 x + 3}$ as a single fraction in its simplest form.

Mark scheme

M1 for factorising $x^{2} - 5 x + 6$ to $(x - 2) (x - 3)$ .
M1 for factorising $x^{2} - 4 x + 3$ to $(x - 1) (x - 3)$ .
M1 for simplifying the second fraction: $\frac{x - 1}{(x - 1) (x - 3)}$ cancels to $\frac{1}{x - 3}$ .
M1 for creating a common denominator $(x - 2) (x - 3)$ and subtracting: $\frac{x + 2}{(x - 2) (x - 3)} - \frac{x - 2}{(x - 2) (x - 3)}$ .
A1 for reaching the final simplified answer $\frac{4}{(x - 2) (x - 3)}$ .

5 steps

Worked example 2 · 4 marks · hard

Show that $\frac{4 x}{x^{2} - 9} - \frac{2}{x - 3}$ can be simplified to the form $\frac{a}{x + b}$ , where $a$ and $b$ are integers.

Mark scheme

M1 for factorising the denominator $x^{2} - 9 = (x - 3) (x + 3)$ .
M1 for finding the common denominator and forming the numerator: $4 x - 2 (x + 3)$ .
M1 for expanding the numerator to $4 x - 2 x - 6 = 2 x - 6$ and factorising it to $2 (x - 3)$ .
A1 for cancelling $(x - 3)$ to achieve $\frac{2}{x + 3}$ , meaning $a = 2$ and $b = 3$ .

4 steps · 1 to fill in

Worked example 3 · 4 marks · medium

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

Mark scheme

M1 for rewriting the division as multiplication by the reciprocal: $\times \frac{x}{4 x^{2} - 1}$ .
M1 for factorising $2 x^{2} - x - 1$ to $(2 x + 1) (x - 1)$ and $x^{2} - x$ to $x (x - 1)$ .
M1 for factorising $4 x^{2} - 1$ to $(2 x - 1) (2 x + 1)$ .
A1 for cancelling all common factors to yield $\frac{1}{2 x - 1}$ .

4 steps

Free account: +5 more practice questions on algebraic fractions with instant feedback, and full worked answers on the free questions. Free, no card. Join free →

Common P3 algebraic fractions mistakes

Cancelling terms, not factors. In $\frac{x + 6}{x + 3}$ nothing cancels: the 3 is part of a sum, not a factor.
Losing a minus sign. $\frac{a}{x} - \frac{b - c}{x} = \frac{a - b + c}{x}$ : the minus applies to the whole second numerator.
Not factorising fully. $x^{2} - 1$ is $(x - 1) (x + 1)$ and $2 x^{2} - 18$ is $2 (x - 3) (x + 3)$ . Hidden common factors are where the marks are.
Dividing without inverting. Turn the second fraction upside down and multiply; dividing tops and bottoms separately rarely works.

Every P3 mistake from the examiner reports, with the paper it came from →

P3 Algebraic Fractions FAQ

What does simplest form mean for an algebraic fraction?

The numerator and denominator have no common factor left. Factorise both fully and cancel; you can leave the answer factorised.

How is this used later in the course?

Simplifying rational expressions is needed for composite and inverse functions in P3 and for partial fractions, integration and the binomial expansion in P4.

Is a calculator allowed in IAL P3?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P3 included. Method marks are for the working you write down, so show it.

How long is the IAL P3 exam?

Each IAL maths unit, P3 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does algebraic fractions come up in P3?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, algebraic fractions was tested in 4 of them (4 questions in all). Any topic can be examined in any series.

Next steps

More P3: Functions · Numerical Methods · Further Differentiation · Further Integration · Trigonometric Functions · Trigonometric Addition Formulae · Exponentials and Logarithms

Your P3 revision kit

Save your P3 answers and track your progressA free account remembers the questions you have done and brings you back to this page.

Join free

Already have an account? Sign in

Other units: IAL P1 · IAL P2 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P3 Algebraic Fractions: exam questions and answers, notes and worked examples

Try it first

P3 Algebraic Fractions notes

Simplifying by cancelling

Multiplying and dividing

Adding and subtracting

Improper fractions and algebraic division

P3 Algebraic Fractions worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 4 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P3 algebraic fractions mistakes

P3 Algebraic Fractions FAQ

What does simplest form mean for an algebraic fraction?

How is this used later in the course?

Is a calculator allowed in IAL P3?

How long is the IAL P3 exam?

How often does algebraic fractions come up in P3?

Next steps

Your P3 revision kit