Factorising – Putting the Brackets Back
Factorising is the reverse of expanding. Instead of removing brackets, you put them back in by finding the highest common factor of all the terms.
Factorising is one of the most practically important skills in all of algebra because it is the standard first move for solving equations set to zero — a quadratic like x² + 5x + 6 = 0 is almost impossible to solve directly, but once factorised into (x+2)(x+3) = 0, the solution falls out immediately from the fact that if two things multiply to zero, one of them must be zero. This exact reasoning (the “zero product property”) underpins the factorising method used throughout the Quadratic Equations page later in this section.
What Is Factorising?
Factorising rewrites an expression as a product of factors. When you factorise, you take out the largest possible factor and write the remaining terms inside brackets.
Step 1 – Find the Highest Common Factor (HCF)
Look at all the terms in the expression. Find the largest number and the highest power of each variable that divides every term.
HCF of 6 and 9 is 3. 6x = 3 times 2x. 9 = 3 times 3. Answer: 3(2x + 3).
HCF of 12 and 8 is 4; HCF of x squared and x is x. So HCF = 4x. Answer: 4x(3x - 2).
HCF = 5ab. 15a squared b = 5ab times 3a. 10ab squared = 5ab times 2b. Answer: 5ab(3a + 2b).
Checking Your Answer
Always expand your answer to check it matches the original expression.
5ab times 3a = 15a squared b. 5ab times 2b = 10ab squared. Total: 15a squared b + 10ab squared. Correct!
Factorising Quadratics: Difference of Two Squares
If an expression looks like a squared - b squared, it always factorises as (a + b)(a - b).
| Expression | Factorised Form |
|---|---|
| x squared - 9 | (x + 3)(x - 3) |
| 4x squared - 25 | (2x + 5)(2x - 5) |
| a squared - 1 | (a + 1)(a - 1) |
Common Mistakes
- Not taking out the full HCF — partially factorising and stopping too soon.
- Leaving a coefficient of 1 outside the bracket and writing nothing: 1(x + 3) should just be (x + 3), but more commonly one forgets to include the correct HCF coefficient.
- Sign errors inside the bracket after factorising.
Key Takeaways
- Factorising is the reverse of expanding — it puts brackets back in.
- Always find the highest common factor first; check by expanding the result.
- Difference of two squares pattern: a squared - b squared = (a + b)(a - b).
Practice: Factorise
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