Mathematics 0580 · syllabus C2.2, E2.2, E2.3
Cambridge IGCSE Mathematics 0580 — Expanding and factorising topical past papers
Collecting like terms, expanding brackets and factorising, and on the Extended route completing the square and simplifying algebraic fractions.
- Questions
- 413
- Years
- 2009–2026
- With mark scheme
- 91%
- Questions 2022–2026
- 154
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Expanding and factorising covers
Algebraic manipulation is rewriting an expression without changing its value. Simplifying collects like terms, so 4a + 2b − a + 5b = 3a + 7b. Expanding multiplies out brackets: a single bracket, as in 3x(2x − 5) = 6x² − 15x, or two brackets, as in (2x + 3)(x − 4) = 2x² − 5x − 12, where every term in the first bracket multiplies every term in the second. Factorising reverses this. Core factorises by taking out common factors, as in 12x² + 8xy = 4x(3x + 2y), and factorising always means factorising fully.
Extended goes further: expanding three brackets, factorising by grouping four terms, the difference of two squares, as in 9x² − 25y² = (3x − 5y)(3x + 5y), perfect squares, quadratics ax² + bx + c including those with a ≠ 1, and cubics with a common factor of x. Completing the square writes x² + 6x + 11 as (x + 3)² + 2, which shows that the lowest point of its graph is (−3, 2). Algebraic fractions are simplified by factorising the top and bottom and cancelling common factors, and they are added, subtracted, multiplied and divided like numerical fractions.
Simplifying, expanding and taking out common factors are Core content, so they can be set on all four papers; the other factorisations, completing the square and algebraic fractions are Extended, examined only on Papers 2 and 4. These skills carry the rest of algebra: factorising solves quadratic equations, completing the square finds turning points, and simplifying is the last step of many answers on equations, sequences, functions and vectors.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C2.2 Algebraic manipulation
- E2.2 Algebraic manipulation
- E2.3 Algebraic fractions
What you should be able to do
- Collect like terms, including terms with powers and products such as ab and a²
- Expand one or two brackets, or three on the Extended route, and simplify
- Factorise fully by taking out the highest common factor
- Factorise quadratics, including ax² + bx + c with a ≠ 1, the difference of two squares and grouping (Extended)
- Write a quadratic in completed-square form (Extended)
- Simplify, add, subtract, multiply and divide algebraic fractions (Extended)
Key ideas
- (a + b)(c + d) = ac + ad + bc + bd: every term multiplies every term
- a² − b² = (a − b)(a + b), and a² + 2ab + b² = (a + b)²
- x² + bx + c = (x + b/2)² − (b/2)² + c
- To factorise ax² + bx + c, find two numbers that multiply to ac and add to b, then split the middle term
- An algebraic fraction can be cancelled only by a factor of the whole top and the whole bottom
How Expanding and factorising has been examined
From 2022 to 2026, Expanding and factorising was on Paper 4 · Calculator (Extended) in 12 of 14 exam series, worth about 6% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 41 | 5% | 14 of 14 |
| Paper 2 · Non-calculator (Extended) | 57 | 8% | 14 of 14 |
| Paper 3 · Calculator (Core) | 23 | 6% | 10 of 14 |
| Paper 4 · Calculator (Extended) | 33 | 6% | 12 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 3.
Filed in Momentum under: Brackets and Simplifying, Factorising.
A worked example
Simplify (x² − 9)/(2x² + 5x − 3).
- The numerator is a difference of two squares: x² − 9 = (x − 3)(x + 3).
- For the denominator, find two numbers that multiply to 2 × (−3) = −6 and add to 5: they are 6 and −1.
- Split the middle term: 2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3).
- Cancel the common factor (x + 3), so the fraction becomes (x − 3)/(2x − 1).
- Check with x = 2: the original gives −5/15 = −1/3, and so does the answer, (2 − 3)/(4 − 1) = −1/3.
Answer: (x − 3)/(2x − 1)
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Squaring a bracket term by term: (x + 5)² is x² + 10x + 25, not x² + 25, because the bracket is multiplied by itself.
- Not factorising fully: 6x² − 9x = 3(2x² − 3x) loses a mark, since x is also a common factor, giving 3x(2x − 3).
- Cancelling terms instead of factors: in (x + 6)/(x + 3) nothing cancels, because 6 and 3 are terms of a sum, not factors of it.
- Losing a sign when expanding with a negative: −2(3x − 4) is −6x + 8, not −6x − 8.
- Adding algebraic fractions by adding the tops and the bottoms: they need a common denominator, just like numerical fractions.
Exam technique
- Check a factorisation by expanding it again; it takes seconds and catches sign errors.
- Before factorising a quadratic, take out any common factor, then look for a difference of two squares or for two numbers with the right product and sum.
- For algebraic fractions, factorise every numerator and denominator first, then cancel or find the common denominator.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.