Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Algebraic Fractions

In one sentence. Simplifying, combining and solving with fractions whose parts are algebra — factorise first, and they behave exactly like number fractions. Higher Tier.

Algebra — one of 8 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with François Viète. Learn Algebraic Fractions, teach it, then defend it against François Viète.

Where this sits in the specification

Specification 4MA1, 2.2 Algebraic manipulation — algebraic fractions (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 2.2 — Algebraic manipulation — algebraic fractions (Higher)

1 · Understand it

What it is

Algebraic fractions follow the same laws as number fractions: cancel common FACTORS, use common denominators to add, cross-multiply to solve. The one new skill is that the factors have to be found first — factorise top and bottom before anything else.

Why it works

You can cancel (x + 3) from top and bottom because it's a factor — a thing the whole expression is multiplied by — exactly as 6/8 cancels a 2. You can never cancel across a + sign: in (x² + 5)/x, the x isn't a factor of the top, so it's locked in place. Factorising is what reveals which cancellations are legal.

When a question is asking for it

'Simplify fully' with quadratics upstairs and down; 'write as a single fraction'; and equations with x in the denominator — multiply through by the denominators and a polynomial equation appears.

How to hold it in your head

Treat every bracket as a sealed unit — a single tile. (x + 3)(x + 2) over (x − 2)(x + 2) is tile·tile over tile·tile, and matching tiles cancel whole. Unsealed terms (things joined by + or −) are not tiles and never cancel.

2 · The mistakes that cost marks

  • Cancelling terms instead of factors: crossing off the x²'s in (x² + 5x + 6)/(x² − 4) ✗. Factorise first; cancel brackets, not letters.
  • Adding by adding tops and bottoms. Common denominator, always — for 2/x + 3/(x+1), it's x(x + 1).
  • Multiplying only some terms when clearing denominators from an equation. EVERY term gets multiplied, both sides.
  • Forgetting that a cancelled bracket still bans a value: (x+3)/(x−2) came from a fraction undefined at x = 2.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Simplify fully (x² + 5x + 6)/(x² − 4).

  1. Factorise top: (x + 2)(x + 3).
  2. Factorise bottom (difference of two squares): (x + 2)(x − 2).
  3. Cancel the shared tile (x + 2).

Answer: (x + 3)/(x − 2)

Write 2/x + 3/(x + 1) as a single fraction.

  1. Common denominator: x(x + 1).
  2. 2/x = 2(x + 1)/[x(x + 1)] and 3/(x + 1) = 3x/[x(x + 1)].
  3. Add the tops: 2(x + 1) + 3x = 5x + 2.

Answer: (5x + 2)/[x(x + 1)]

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Simplify 4x² ÷ 2x (for x ≠ 0).
  2. Grade 7 — Simplify (x² − 9)/(x + 3) (for x ≠ −3).
  3. Grade 8 — Write 2/x + 3/(x + 1) as a single fraction. (Type it like (5x+2)/(x(x+1)).)
  4. Grade 9 — Simplify fully (2x² − 5x − 3)/(x² − 9).
  5. 9+ — Solve 5/x + 2 = 3/(2x). Give your answer as a fraction.

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Algebraic Fractions, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Algebraic Fractions works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Algebraic Fractions against François Viète — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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