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Factorising is the reverse of expanding: where expanding turns 3(x + 4) into 3x + 12, factorising turns 3x + 12 back into 3(x + 4). 12 questions answered.
Mastering Basic Algebraic Processes is part of The basic processes of algebra in GCSE AQA Higher Further Maths.
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Factorising is the reverse of expanding: where expanding turns 3(x + 4) into 3x + 12, factorising turns 3x + 12 back into 3(x + 4). 12 questions answered.
Use this lesson to stay inside The basic processes of algebra while connecting the detail back to the wider Algebra topic.
Read the core explanations from Mastering Basic Algebraic Processes before testing yourself from memory.
Numbers follow predictable patterns — and so does algebra. The three laws you are about to meet are not arbitrary rules; they are the reason you can rearrange, expand, and simplify any expression without changing its value.
Every manipulation you perform in algebra — collecting like terms, expanding brackets, factorising — rests on these three foundations. Get them right here, and the rest of GCSE algebra becomes far more manageable.
You already use this law without realising it. Writing $3x + 5$ or $5 + 3x$ gives the same expression — the commutative law guarantees that. The same holds for multiplication: $4 \times x = x \times 4$.
This law tells you that when adding or multiplying a chain of terms, the brackets can move freely. So $(2x + 3x) + 4x = 2x + (3x + 4x) = 9x$ — the grouping is your choice.
The distributive law is the most heavily tested of the three. Every time you expand a bracket, you are applying it. Notice that the term outside the bracket multiplies every term inside — not just the first one.
Before expanding brackets, you need to be confident collecting like terms — this is the most fundamental algebraic process. Like terms share the same variable and the same power: $3x$ and $7x$ are like terms, but $3x$ and $3x^2$ are not.
The commutative and associative laws together allow you to reorder and regroup terms so that like terms sit next to each other, ready to be combined. Think of it as sorting: group the $x$ terms, group the $x^2$ terms, group the constants.
Worked example: Collecting Like Terms
Simplify $5x^2 + 3x - 2x^2 + 7 - x + 4$.
Answer: $3x^2 + 2x + 11$
Expanding brackets is the distributive law in action. The term outside the bracket multiplies every single term inside — this is the step where most errors occur.
Worked example: Expanding a Single Bracket
Expand and simplify $3(2x + 5) - 4x$.
Answer: $2x + 15$
Expanding two brackets uses the same distributive law, applied twice. Every term in the first bracket multiplies every term in the second bracket. A reliable method is to use FOIL: First, Outer, Inner, Last.
Worked example: Expanding Double Brackets
Expand and simplify $(x + 3)(x - 5)$.
Answer: $x^2 - 2x - 15$
Factorising is the reverse of expanding. Where expanding turns $3(x + 4)$ into $3x + 12$, factorising turns $3x + 12$ back into $3(x + 4)$. The key question to ask is: what is the highest common factor of all the terms?
The highest common factor (HCF) is the largest number and the highest power of any shared variable that divides into every term. Once you have identified it, place it outside the bracket and divide each term by it to find what goes inside.
Worked example: Factorising by Common Factor
Factorise $6x^2 + 9x$ fully.
Answer: $3x(2x + 3)$
The errors that cost marks in algebra questions are almost always the same ones. Knowing them in advance means you can check for them deliberately rather than hoping for the best.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Factorising is the reverse of expanding: where expanding turns 3(x + 4) into 3x + 12, factorising turns 3x + 12 back into 3(x + 4). The highest common factor of all the terms is placed outside the bracket.
FOIL is a method for expanding two brackets: multiply the First terms, then the Outer terms, then the Inner terms, then the Last terms. For (x + 3)(x − 5), this gives x² − 5x + 3x − 15 = x² − 2x − 15.
A constant is a plain number with no variable. When simplifying 5x² + 3x − 2x² + 7 − x + 4, the constants 7 and 4 combine to 11, giving 3x² + 2x + 11.
The distributive law states that multiplication distributes over addition and subtraction: a(b + c) = ab + ac and a(b − c) = ab − ac. It is the engine behind expanding brackets, because the term outside the bracket multiplies every term inside, not just the first one.
The commutative law states that for addition and multiplication, order does not matter: a + b = b + a and a × b = b × a. It does not apply to subtraction or division. It is why 3x + 5 and 5 + 3x are the same expression.
The associative law states that for addition and multiplication, regrouping does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). For example, (2x + 3x) + 4x = 2x + (3x + 4x) = 9x.
To factorise by taking out a common factor, find the highest common factor (HCF) of all the terms, place it outside the bracket and divide each term by it. For 6x² + 9x, the HCF is 3x, so 6x² + 9x = 3x(2x + 3).
The highest common factor of algebraic terms is the largest number and the highest power of any shared variable that divides into every term. For 6x² + 9x, the HCF of 6 and 9 is 3 and both terms contain x, so the HCF is 3x.
The commutative law does not apply to subtraction or division. It holds only for addition and multiplication, where a + b = b + a and a × b = b × a.
A single bracket is expanded by multiplying the term outside by every term inside, then collecting like terms. For 3(2x + 5) − 4x, this gives 6x + 15 − 4x, which simplifies to 2x + 15.
Factorising is checked by expanding the answer back out. For example, 3x(2x + 3) expands to 3x × 2x + 3x × 3 = 6x² + 9x, the expression that was factorised.
The most common mistake when expanding brackets is multiplying only the first term inside by the term outside. The distributive law means the term outside multiplies every single term inside the bracket.