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The essential algebra skills that underpin every IGCSE Maths paper. Step-by-step guide with FOIL, DOTS and AC method worked examples for grades 5-9.
Expanding brackets means multiplying out expressions such as 3(x + 4) or (x + 2)(x - 5) to remove the brackets. Factorising is the reverse process: rewriting an expression as a product of its factors. Both skills are tested heavily across Paper 1 and Paper 2 and form the foundation for solving equations, simplifying algebraic fractions, and completing the square.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
Expanding brackets means multiplying every term inside the bracket by the term (or expression) outside. The goal is to remove all brackets and write the result as a simplified polynomial.
Multiply the term outside by each term inside. For example, 3(2x + 5) = 6x + 15. Be especially careful when the outside term is negative: −2(x − 4) = −2x + 8, not −2x − 8.
To expand two binomials such as (x + a)(x + b), use the FOIL mnemonic:
Then collect like terms. For example, (x + 3)(x − 7) = x² − 7x + 3x − 21 = x² − 4x − 21.
Expand two brackets first using FOIL, then multiply the result by the third bracket term by term. Always simplify fully after each stage to reduce errors.
Factorising is the reverse of expanding. You rewrite an expression as a product of simpler expressions. There are four key techniques tested at IGCSE level:
Find the highest common factor (HCF) of all terms and place it outside a bracket. For example, 6x² + 15x = 3x(2x + 5). Always check for both numerical and algebraic common factors.
The identity a² − b² = (a − b)(a + b) applies whenever you have one perfect square subtracted from another. For example, 25x² − 49 = (5x − 7)(5x + 7). Always look for a common factor first: 2x² − 50 = 2(x² − 25) = 2(x − 5)(x + 5).
For x² + bx + c, find two numbers that multiply to c and add to b. For example, x² + 5x + 6: the pair is +2 and +3, so the factorisation is (x + 2)(x + 3).
For ax² + bx + c where a > 1, multiply a × c to find the product ac. Find two numbers that multiply to ac and add to b. Split the middle term using these numbers, then factorise by grouping in pairs.
Algebraic identities are equations that are true for all values of the variable. Recognising these patterns instantly saves valuable exam time.
| Identity | Expanded Form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| a² − b² | (a − b)(a + b) |
The perfect square identities are particularly useful when completing the square. The difference of two squares is essential for simplifying algebraic fractions and solving equations efficiently.
Expand and simplify: (x + 2)(x − 3)(2x + 1)
Step 1 (Expand first two brackets): (x + 2)(x − 3) = x² − 3x + 2x − 6 = x² − x − 6
Step 2 (Multiply by third bracket): (x² − x − 6)(2x + 1)
Step 3 (Expand term by term): = 2x³ + x² − 2x² − x − 12x − 6
Step 4 (Collect like terms): = 2x³ − x² − 13x − 6
Answer: 2x³ − x² − 13x − 6
Factorise completely: 4x² − 81y²
Step 1 (Recognise DOTS): 4x² = (2x)² and 81y² = (9y)²
Step 2 (Apply a² − b² = (a − b)(a + b)):
= (2x − 9y)(2x + 9y)
Answer: (2x − 9y)(2x + 9y)
Factorise: 3x² + 10x − 8
Step 1 (Find ac): a × c = 3 × (−8) = −24
Step 2 (Find factor pair): Two numbers that multiply to −24 and add to +10: that is +12 and −2
Step 3 (Split the middle term): 3x² + 12x − 2x − 8
Step 4 (Factorise by grouping): 3x(x + 4) − 2(x + 4)
Step 5 (Take out common bracket): (3x − 2)(x + 4)
Answer: (3x − 2)(x + 4)
Sign trap when expanding negatives: −3(x − 4) = −3x + 12, not −3x − 12. The negative multiplies both terms, flipping the sign of −4 to +12.
Incomplete factorisation: Students write 2x² − 50 = 2(x² − 25) and stop. The expression inside is DOTS, so the full answer is 2(x − 5)(x + 5). Always check if you can factorise further.
Not checking by re-expanding: The fastest way to verify your factorisation is to expand it back out. If you get the original expression, your answer is correct. This takes seconds and can save crucial marks.
Confusing signs in DOTS: a² − b² = (a − b)(a + b), not (a − b)(a − b). One bracket has a minus, the other has a plus.
Factorise fully: If the question says "factorise completely", examiners expect every possible factor to be extracted. Check for common factors first, then DOTS or quadratic methods.
Always check by expanding: After factorising, expand your answer mentally or on scrap paper. If it matches the original, you have full marks.
Look for the DOTS pattern: Before attempting other methods, scan for the difference of two squares. It is the fastest factorisation technique and is frequently tested.
Show all working: Method marks are awarded for correct intermediate steps even if the final answer contains an arithmetic error. Write every line clearly.
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