Is your child struggling with factorisation in Secondary Math? You’re not alone. Many students find factorisation confusing at first, especially when it comes to recognising which method to use and when. But here’s the good news: once you understand the patterns and techniques, factorisation becomes much more straightforward.
The truth is, factorisation is everywhere in secondary math. From solving equations to simplifying fractions and graphing curves, this skill appears again and again. That’s why mastering it now will make every other algebra topic easier later on.
In this guide, we’ll walk you through everything you need to know about factorisation, from the basic concept to all four main methods. Whether your child is in Sec 1 or preparing for O Levels, this article will help them build confidence and accuracy in factorisation.
Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
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What is Factorisation?
Factorisation is the process of rewriting an expression as a product of two or more simpler expressions called factors. Think of it as doing the reverse of expansion. Instead of opening up brackets, you’re breaking down a complicated expression into simpler pieces that multiply together.
Here’s a simple example to show how expansion and factorisation work in opposite directions:
Expansion:
In the above example, you can see how expansion and factorisation work in opposite directions.
- When you expand 6(x + 2), you multiply 6 by each term inside the bracket to get 6x + 12.
- When you factorise, you do the reverse. You look at 6x + 12 and notice that both terms share a common factor of 6. By “taking out the 6”, you rewrite the expression as a product → 6(x + 2).
This shows that factorisation is simply turning a long expression back into a product of simpler factors.
Factorisation in algebra works very much like breaking whole numbers into prime factors which is taught in secondary 1. For example, the number 12 can be written as 12 = 2 × 2 × 3. These are the building blocks that multiply to give you 12.
Why is Factorisation Important?
Factorisation is a core skill in secondary mathematics because you will use it in many topics, not just in Algebra. Once you know how to break expressions into factors, you will find that many problems become much easier to solve.
- It helps you solve equations
There are many equations that you can only solve after factorising. When an equation is written in factored form, you can set each factor to zero and solve step by step. This reduces the amount of working you need to do and helps you find solutions more quickly.
Example [secondary 2 level]
Solve x2 – 5x = – 6.
Solution:
Given x2 – 5x = 6 ⇒ x2 – 5x + 6 =
[Factorise the left–hand–side]
(x – 2)(x – 3) = 0
[See that the product gives us a final value of 0. Therefore either one of the expressions on the left must have a value of 0.]
x – 2 = 0 or x – 3 = 0
x = 2 x = 3
Ans: x = 2 or x = 3
- It helps you to write expressions in a more concise manner
Some expressions look long or complicated, but factorisation lets you rewrite them in a much clearer way. This helps you understand the structure of the expression instead of getting lost in too many terms.
Factorisation also allows us to cancel out common factors (in algebraic fractions) that are not obvious.
Example [secondary 2]
Factorise 10ab + 4bc – 15ak – 6ck completely.
Solution:
10ab + 4bc – 15ak – 6ck = 2b(5a + 2c) – 3k(5a+ 2c)
= (5a + 2c)(2b – 3k) (ans)
Example [secondary 2]
Solution:
- It prepares you for upper-secondary maths
Almost all topics in secondary math (such as quadratic functions, inequalities, graphs of curves, indices etc) will involve factorisation. If you become confident in factorising now, the later chapters will feel a lot easier, and you will make less careless mistakes.
- It helps you check your work
As mentioned earlier, we can treat factorisation as the reverse of expansion. Hence we can use factorisation to check if our answers make sense after expansion.
Example [secondary 1]
Expand 8x(2 + y).
Incorrect solution: Check:
8x(2 + y) = 16x + 8y 16x + 8y = 8(2x + y)
≠ 8x(2 + y) ✗
See that when we factorise our initial answer, we do not obtain the original expression of 8x(2 + y). Hence, we know this answer must be incorrect.
Correct solution: Check:
8x(2 + y) = 16x + 8xy 16x + 8xy = 8x(2 + y) ✓
The 4 Main Types of Factorisation
When you factorise, your goal is always the same: rewrite an expression as a product of simpler factors. However, there are four main methods you will use in secondary school. Once you know which method fits the expression, the question becomes much easier.
1. Common Factors
This is the easiest and most important method. You look for the highest common factor (HCF) in all the terms. The GCF can be a number, a variable, or a combination of both.
– Identify the HCF
You compare all the terms and pick out the largest shared number and all the shared letters.
For example:
12x² + 8x
The HCF of 12 and 8 is 4.
Both terms contain the variable x.
∴ HCF = 4x.
So, 12x² + 8x = 4x(3x + 2)
– Typical mistakes to avoid
- Not factorising completely
Example: 25xy + 10x = 5(5xy + 2x) [This expression is factorised, but not factorised completely]
25xy + 10x = 5x(5y + 2) ✓
- Does not recognise that factoring out the factor itself leaves 1 behind
Example: a²b + 12ac + a = a(ab + 12c) ✗
a²b + 12ac + a = a(ab + 12c + 1) ✓
- Leaving out brackets or placing brackets at wrong places
Example: 25xy – 10x + 5 = 5(5xy – 2x) + 1 ✗
25xy – 10x + 5 = 5(5xy – 2x + 1) ✓
- Forgetting to divide each term correctly after taking out the HCF
Example: 6x3y– 3xy2 = 3xy(2x2 – 3y) ✗
6x3y – 3xy2 = 3xy(2x2 – y) ✓
To check your answer, expand your brackets. If you get back the original expression, your factorisation is correct.
2. Grouping Method
You use grouping when there is no single common factor across all terms, but you can group them into pairs that share a factor.
When to use it
You usually use grouping when there are four terms:
ax + ay + bx + by
You group the first two and the last two terms.
Patterns to look for
- The first pair shares a common factor.
- The second pair shares a common factor.
- After factorising both pairs, you should see a repeated bracket. Factorise out this repeated bracket.
Example
2x + 6y + xz + 3zy [See that 2 is common factor for first pair and z is common factor for second pair]
= 2(x + 3y) + z(x + 3y) [See that (x + 3y) is repeated. Factorise this out. Many students have difficulty with this. Recall this concept:
2a + za = a(2 + z). Here, a is simply (x + 3y) ]
= (x + 3y)(2 + z)
– Typical mistakes to avoid
- Did not pull out negative terms when needed
Example:
3x – 3y + bx – by = 3(x – y) – b(–x + y) [Pulling out “–b” instead of “b”. This does not give us any repeated factors.]
2p – 2q – np + nq = 2(p – q) – n(p – q)
= (p – q)(2 – n) ✓
- Pull out negative terms when not needed
Example:
3x – 3y + bx – by = 3(x – y) – b(–x + y) [Pulling out “–b” instead of “b”. This does not give us any repeated factors.]
3x – 3y + bx – by = 3(x – y) + b(x – y)
= (x – y)(3 + b) ✓
- Grouping the wrong way and ending with unmatched brackets
Example:
2kp2 + 6 – 3p2 – 4k = 2(kp2 + 3) – (3p2 + 4k) [pulling out “2” and “–1” will not give us repeated factors and we become “stuck”]
2kp2 + 6 – 3p2 – 4k = 2kp2 – 3p2 + 6 – 4k [switch the centre terms]
= p2(2k – 3) –2(2k – 3) [pull out ” p2 ” and “–2”]
= (2k – 3)(p2 – 2) ✓
3. Quadratic expressions (ax² + bx + c)
There are two main cases you will learn.
Case 1: x² + bx + c
When a = 1, the coefficient of x² = 1:
- express c as a product of 2 integers.
- check if the two integers add up to b.
Example 1
Factorise x² + 7x + 10.
Try: 10 = 5 × 2 Check: 5 + 2 = 7 ✓
x² + 7x + 10 = (x + 5)(x + 2) (ans)
Factorise x² + 3x – 10
Try: –10 = (–5) × 2 But –5 + 2 ≠ 3 ✗
Try: –10 = 5 × (–2) Check: 5 + (–2) = 3 ✓
x² + 3x – 10 = (x + 5)(x – 2) (ans)
Case 2: ax² + bx + c
We apply the “cross-multiplication frame” method to factorise as this involves more guess and check. The frame will help us to minimise careless mistakes.
To use to factorise 3x2 + 14x – 5 using the frame, follow the following steps.
Step 1: Write out the given quadratic expression in the third row in the order of ax2 + c + bx.
Step 2: Factorise the quadratic term ax2 , into two linear factors. [ 3x2 is a result of (3x)(x) ]
Write out the factors in the first column for rows 1 and 2.
Step 3: Factorise the constant term – 5, into a product of two numbers. [–5 is a result of (–1)(5) ]
Write out the factors in the second column for rows 1 and 2.
Step 4: Cross multiply the factors and write the products in the third column. Add up the products in the third column and check that the sum is indeed +14x. If not, we need to try other combinations for ax2 and/or c.
Based on the above set-up, we verify that –x + 15x is indeed +14x. Hence
3x2 + 14x – 5 = (3x – 1)(x + 5)
One example where we must “guess and check” again is shown below. Assume we have actually reversed the positions of –1 and 5.
Observe that we will obtain 5x and –3x in the third column, which do not add up to +14x.
Further Example 1
x2 + 8x + 7 = (x + 1)(x + 7)
The above shows that this frame can also be used even when a = 1.
Further Example 2
3x2 – 10x + 7
= (3x – 7)(x – 1)
Further Example 3 [quadratic expression with 2 variables]
5x2 – 7xy – 6y2
= (5x + 3y)(x – 2y)
In Further Example 3, the term with 2 variables (xy) will be treated as bx and placed in the third row of the third column.
4. Special Identities
These are expressions with fixed patterns. Once you recognise the pattern instantly, factorisation becomes very fast.
Difference of squares identity:
a² − b² = (a − b)(a + b)
Example:
9x² − 16 = (3x − 4)(3x + 4)
Perfect square identities:
(i) a² + 2ab + b² = (a + b)²
(ii) a² − 2ab + b² = (a − b)²
Examples:
(i) x² + 12x + 36 = (x + 6)²
(ii) 9p2 – 12p + 4 = (3p – 2)²
Note that the factorisation of examples (i) and (ii) can also be done using multiplication frame.
Sum and difference of cubes [only applicable for Additional Mathematics]
(i) a³ + b³ = (a + b)(a² − ab + b²)
(ii) a³ − b³ = (a − b)(a² + ab + b²)
Example:
x³ + 8 = x³ + 2³ = (x + 2)(x² − 2x + 4)
How to spot patterns instantly
Examples of patterns you can spot instantly:
(i) Recognise squares such as 4, 9, 16x2 or 25y2 etc
Examples:
- 6x² – 25 → this is 4²x² – 5² → difference of squares
- x² + 16x + 64 → this is x² + 2(8)x + 8² → perfect square → (x + 8)²
- 49a² – 81b² → this is 7²a² – 9²b² → difference of squares → (7a – 9b)(7a + 9b)
(ii) Recognise cubes such as 8, 27, or 64p3 etc
Examples:
- x³ – 27 → this is x³ – 3³ → difference of cubes → (x – 3)(x² + 3x + 9)
- 64 + y³ → this is 4³ + y³ → sum of cubes → (4 + y)(16 – 4y + y²)
(iii) See that the middle term being exactly twice the product of the square roots (perfect square case)
You check if 2(√ax2)(√c) = bx in ax² + bx + c.
Examples:
- x² + 12x + 36 → 2(√x2)(√36) = 12x → yes → (x + 6)²
- 4x² – 20x + 25 → 2(√4x2)(√25) = 20x → yes → (2x – 5)²
(iv) See a minus sign between two squares (difference of squares)
Examples:
- x² – 49 → x² – 7² → (x – 7)(x + 7)
- 25a² – b² → (5a – b)(5a + b)
- 4m² – 81n² → (2m – 9n)(2m + 9n)
How to Know Which Method to Use
When you face any algebraic expression, your first challenge is choosing the correct method of factorisation. The steps below act like a decision checklist to guide you.
(i) Look for a common factor first.
You must do this even if the question looks like a quadratic or identity.
Example:
6x² + 12x
There is a common factor of 6x.
Factorised:
6x(x + 2)
Not checking this is one of the biggest mistakes students make.
(ii) If there are four terms, try grouping.
If you see four terms, or two obvious pairs, try grouping.
Example:
xy + 3y + 2x + 6 = y(x + 3) + 2(x + 3)
= (x + 3)(y + 2)
(iii) If there are three terms and the highest power is x², try quadratic factorisation.
Example:
x² + 7x + 10
Numbers multiply to 10 and add to 7 → 5 and 2.
(x + 5)(x + 2)
*Since 2(√x2)(√10) ≠ 7x, we know that this does not involve special quadratic identities.
(iv) If the expression matches a known pattern, use special identities. Look for perfect squares, difference of squares, or cube patterns.
Examples:
25 – x² This matches a² – b² → (a – b)(a + b).
5² – x² → (5 – x)(5 + x)
4p2 – 12p +9 Check: 2(√4p2)(√9) = 12p, so we can apply a special quadratic identity. Although you can also apply the cross–multiplication frame immediately without checking.
4p2 – 12p + 9 = (2p – 3)2
Common Mistakes to Avoid
These are errors that students make very often. Examples are included so you can recognise and avoid them.
- Forgetting to factor out 1
Every expression can be factored out by 1, but the point is to remember that sometimes nothing else is common.
Example mistake:
np – nq + p – q = n(p – q) + p – q ✗
np – nq + p – q = n(p – q) + 1(p – q)
= (p – q)(n +1) ✓
- Treating expressions like equations
Some students try to “solve” instead of factorising.
Example mistake:
Factorise x² + 9x + 20.
x² + 9x + 20 = 0
(x + 5)(x + 4) = 0
x = – 5 or x = –4 ✗
x² + 9x + 20 = (x + 5)(x + 4) ✓
- Expanding wrongly after factorising
You should always check your answer by expanding.
Example mistake:
(x – 3)(x – 2)
A student might expand to get
x² – 6
Correct expansion is
x² – 5x + 6
- Sign mistakes
Positive and negative signs are the most common error.
Example:
x² – x – 12
Integer numbers must multiply to –12 and add to –1.
Correct factorisation:
(x – 4)(x + 3)
A common mistake is
x² – x – 12 = (x + 4)(x – 3) ✗ or x² – x – 12 = (x + 6)(x – 2) ✗
- ▪Assuming all quadratics factorise
Not every quadratic has neat whole-number factors.
Example:
x² – 2x + 7
No pair of numbers multiply to 7 and add to –2.
This quadratic cannot be factorised using integers.
Students should recognise this quickly to avoid wasting time.
Conclusion
Factorisation may seem difficult when you first start learning it, but it becomes much easier once you understand the main ideas and recognise the patterns. Every method you learnt in this guide builds towards the same goal: rewriting an expression as a product of simpler parts.
When you follow the checklist, look for common factors first, and match each expression to the correct method, you will find that algebra problems become far more manageable.
With enough practice, you will be able to spot useful patterns instantly, avoid common mistakes, and factorise with confidence. These skills will not only help you in algebra but also support many topics you will learn later in secondary school, especially equations, graphs, inequalities, and more complex expressions.
Keep practising, check your work using expansion, and you will steadily become more accurate and faster. Factorisation is a skill that rewards consistency, and the more you use it, the more natural it will feel.
Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
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