Factorising: the opposite of expanding; it means "putting into brackets".
Factors: parts of a number that divide exactly into it. For example, 5 divides into 10.
Prime number: a number that has exactly two factors — 1 and itself.
Prime factors: factors of a number that are prime.
Highest Common Factor aka HCF: The highest factor of multiple numbers that divides into them
Factorising single brackets
Suppose you have a linear expression, \(2x + 2\):
To fully factorise this expression, list all the prime numbers:
Use the list of prime numbers to break each term down and find the highest number that divides all terms — this might be a prime or a product of primes.
In this example, the largest prime number is 2.
Next, place the highest common factor outside the bracket, and divide each term in the original expression by it to go inside the bracket. Expanding should give you the original expression back — this helps you check your work.
So, the factorised expression is \(2(x + 1)\). Remember that there is an invisible 1 next to x.
Remember that when you fully factorise an expression, there is at least one prime outside or inside the bracket, that's when you know you've done it correctly.
So follow LION:
L – LIST the prime factors and find the HCF.
I – Insert the divided terms INSIDE the brackets.
O – OUTSIDE the bracket goes the HCF.
N – NEXT, expand to check your answer.
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Remember, sometimes the factor can be a variable. In \(4x + 8xy\), the highest common factor is \(4x\), so it becomes \(4x(1 + 2y)\).
When factoring an \(x^2\) it's the same principle, you just take an x out of the expression if x is a common factor.
Two Term Factorisation Quiz
Score: 0 / 20
Factorising Multiple Terms in an Expression
It's the same principle. Use LION. Find the highest number that fits into each number/term in the expression.
Remember that you can factor out terms as well as numbers and constants. If you do not know what these mean Look here
Three-Term Factorisation Quiz
Score: 0/10
Extra resources
Click here for questions on the topic and here for the answers.
Summary
Factorising = Putting into brackets (the reverse of expanding).
Factors = Numbers that divide exactly into another (e.g., 5 divides into 10).
Prime Number = Has exactly 2 factors: 1 and itself.
Prime Factors = Prime numbers that multiply to give another number.
Highest Common Factor (HCF) = The biggest number that divides all terms.
Factorising Single Brackets
Identify the HCF using a list of prime numbers.
Pull out the HCF and put it outside the brackets.
Divide each term by the HCF and place the result inside the brackets.
Check your work by expanding back out.
Example: \(2x + 2 = 2(x + 1)\)
LION Strategy
L – LIST the prime factors and find the HCF.
I – Insert the divided terms INSIDE the brackets.
O – OUTSIDE the bracket goes the HCF.
N – NEXT, expand to check your answer.
Additional Tips
Factors can include variables (e.g., \(4x + 8xy = 4x(1 + 2y)\)).
When fully factorised, at least one prime or variable will be outside or inside the brackets.
Use LION for multi-term expressions — always find what’s common.
