Factorisation in Maths: How to Factorise Any Expression
Factorisation means writing an expression as things multiplied together. It is expanding brackets run backwards: instead of removing brackets, you put them back.
Check it by expanding back: 3 × 2x = 6x, and 3 × 3 = 9. You get what you started with, so the factorising is right. That check works every single time, which makes this one of the few topics in maths where you can always mark your own answer.
There are really only five things you will ever be asked to factorise, and almost all the marks lost on this topic come from two places: picking the wrong method, and stopping too early.
| What you see | What it is called | Example |
|---|---|---|
| Every term shares something | Common factor | 12x2 + 8x = 4x(3x + 2) |
| Square − square, two terms | Difference of two squares | x2 − 25 = (x + 5)(x − 5) |
| Three terms, starts x2 | Quadratic, a = 1 | x2 + 7x + 12 = (x + 3)(x + 4) |
| Three terms, a number in front of x2 | Quadratic, a > 1 | 2x2 + 7x + 3 = (2x + 1)(x + 3) |
| Four terms | Grouping | xy + 3x + 2y + 6 = (x + 2)(y + 3) |
This page works through all five, then deals with the thing almost no revision page explains: what “factorise fully” actually asks you to do, and why an answer can be completely correct algebra and still score zero.
What factorising actually is
You already know how to expand brackets. Factorising is the same journey in the other direction.
“Factorise” and “factorisation” both come from the word factor, meaning something that divides into a number exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12, and 12 can be written as 3 × 4. Factorising an algebraic expression is the same idea: write it as things multiplied together.
Factorising means turning a sum into a product. Plus signs become times signs.
That is also why it matters far beyond this topic. A sum has no useful zeros, but a product does: if two things multiply to give zero, one of them must be zero. That single fact is what lets you solve quadratic equations, sketch curves, simplify fractions and cancel algebraic terms, and it is covered at the end of this page.
Which method? The thirty-second decision
Students do not usually lose marks because they cannot factorise. They lose them because they pick the wrong method, or because they stop one step early. This chart fixes both.
Scroll sideways to see the whole drawing
Step 1 is the one people skip. Before anything clever, look at whether every term shares a factor — a number, a letter, or both. Taking it out first makes everything that follows easier, and an answer that leaves it in is usually marked wrong even when the algebra is correct.
Type 1: taking out a common factor
This is the one you will use most, and it is the first move on almost every question. You are looking for the highest common factor of all the terms — the biggest thing that divides into every one of them.
- Find the HCF of the numbers. The biggest number that goes into all of them.
- Find the letters they all share, taking the lowest power of each. If one term has x3 and another has x, you can only take out x.
- Write that outside a bracket, then divide every original term by it to fill the bracket in.
- Expand it back to check you get the question.
Factorise 12x2 + 8x
Numbers: HCF of 12 and 8 is 4
Letters: both terms have x. Lowest power is x
so the HCF is 4x
Divide each term by 4x
12x² ÷ 4x = 3x
8x ÷ 4x = 2
12x² + 8x = 4x(3x + 2)
Check: 4x × 3x = 12x², 4x × 2 = 8x
Factorise 15a3b − 20a2b2
Numbers: HCF of 15 and 20 is 5
a: lowest power is a². b: lowest power is b
so the HCF is 5a²b
Divide each term by 5a²b
15a³b ÷ 5a²b = 3a
20a²b² ÷ 5a²b = 4b
= 5a²b(3a − 4b)
Watch the minus. It stays with the 4b inside the bracket. A very common error is to take out 5a2b and then write a plus.
Writing 12x2 + 8x = 2(6x2 + 4x) is true, but it is not factorised fully — there is still a 2 and an x to come out. If the question says “fully”, that answer scores nothing. Always take the highest common factor, not just a common factor.
Type 2: the difference of two squares
When you have two terms, one square minus another square, there is a shortcut that takes about four seconds.
Three conditions, all of which must hold:
- Exactly two terms. Three terms is a quadratic, not this.
- Both are perfect squares. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and any even power of a letter.
- A minus between them. x2 + 25 does not factorise at all at this level. Say so and move on — that is the correct answer, not a sign you have gone wrong.
Example
x2 − 25
√(x2) = x, √25 = 5
= (x + 5)(x − 5)
With a coefficient
9x2 − 16
√(9x2) = 3x, √16 = 4
= (3x + 4)(3x − 4)
The order of the brackets does not matter — (x − 5)(x + 5) is equally correct, because multiplication can be done either way round.
2x2 − 50 does not look like a difference of two squares, because 2 and 50 are not squares. Take out the common factor of 2 first and it appears: 2(x2 − 25) = 2(x + 5)(x − 5). This is exactly why step 1 is “take out the common factor, always”.
Type 3: quadratics where a = 1
Three terms, starting with a plain x2 with no number in front. These go into two brackets, and the whole job is finding two numbers.
In x2 + bx + c, those two numbers go straight into the brackets.
Factorise x2 + 7x + 12
Need two numbers: multiply to 12, add to 7
1 × 12 = 12, 1 + 12 = 13 no
2 × 6 = 12, 2 + 6 = 8 no
3 × 4 = 12, 3 + 4 = 7 YES
x² + 7x + 12 = (x + 3)(x + 4)
Getting the signs right
Signs cause more lost marks here than the arithmetic does. Read the signs of b and c and you can predict the answer before you start.
| Last sign (c) | Middle sign (b) | The two numbers are | Example |
|---|---|---|---|
| plus | plus | both positive | x2 + 7x + 12 = (x+3)(x+4) |
| plus | minus | both negative | x2 − 5x + 6 = (x−2)(x−3) |
| minus | either | one of each. The bigger one takes the sign of b | x2 + 2x − 15 = (x+5)(x−3) |
Check that last row against the rule. The numbers are +5 and −3: they multiply to −15 and add to +2. The bigger number, 5, carries the plus, because b is positive.
Type 4: quadratics where a > 1
This is the one students dread, because guessing stops working as soon as there is a number in front of the x2. There is a method that never guesses. It is called splitting the middle term, or the AC method.
- Multiply a by c — the first number times the last number.
- Find two numbers that multiply to ac and add to b. Exactly the same hunt as before, just with a different target.
- Split the middle term into those two. You now have four terms instead of three.
- Factorise in pairs, and the same bracket will appear twice. Take it out.
Factorise 2x2 + 7x + 3
Step 1 — a × c = 2 × 3 = 6
Step 2 — two numbers: multiply to 6, add to 7
6 and 1 (6 × 1 = 6, 6 + 1 = 7)
Step 3 — split the 7x into 6x + x
2x² + 6x + x + 3
Step 4 — factorise each pair
2x(x + 3) + 1(x + 3)
(x + 3) appears twice. Take it out
= (2x + 1)(x + 3)
Check: 2x×x + 2x×3 + 1×x + 1×3 = 2x² + 7x + 3
Factorise 6x2 + 11x − 10
Step 1 — a × c = 6 × (−10) = −60
Step 2 — multiply to −60, add to 11
15 and −4 (15 × −4 = −60, 15 − 4 = 11)
Step 3 — split the 11x
6x² + 15x − 4x − 10
Step 4 — factorise each pair
3x(2x + 5) − 2(2x + 5)
= (3x − 2)(2x + 5)
The second pair is −4x − 10, and taking out −2 rather than 2 is what makes the bracket match. If your two brackets do not come out identical, try taking out the negative instead.
The order you split the middle term does not matter. Splitting 6x2 − 4x + 15x − 10 gives 2x(3x − 2) + 5(3x − 2) = (2x + 5)(3x − 2), which is the same answer with the brackets swapped.
Type 5: factorising by grouping
Four terms almost always means grouping. You will have just used this as step 4 of the splitting method, so it is already familiar.
Factorise xy + 3x + 2y + 6
Split into two pairs
(xy + 3x) + (2y + 6)
Factorise each pair separately
x(y + 3) + 2(y + 3)
(y + 3) appears twice. Take it out
= (x + 2)(y + 3)
The signal that it is working is the identical bracket appearing twice. If your two brackets differ, re-pair the terms in a different order, or take out a negative from the second pair.
What “factorise fully” really means
This word appears on exam papers constantly and almost no revision page explains it. “Fully” means keep going until nothing inside any bracket can be factorised again.
It is possible to write algebra that is completely true and still score zero, because you stopped one step early. That is what the word is there to catch.
True, but not fully factorised
4x2 − 36 = (2x + 6)(2x − 6)
Expand it and you do get 4x2 − 36. But both brackets still have a 2 in them.
Fully factorised
4x2 − 36 = 4(x + 3)(x − 3)
Common factor out first, then the difference of two squares. Nothing left to take.
Both lines are mathematically true. Only the second one gets the mark.
The two-pass habit
Do everything twice, deliberately:
- First pass — take out the highest common factor of the whole expression.
- Second pass — look inside what is left. Is it a difference of two squares? A quadratic? If so, factorise that too.
- Third look — check every bracket you have written. Can any of them go further? Usually not, but checking is what the word “fully” is asking for.
Factorise fully 5x3 − 45x
Pass 1 — common factor. Both terms have 5 and x
5x(x² − 9)
Pass 2 — is x² − 9 anything? Square − square, yes
= 5x(x + 3)(x − 3)
Pass 3 — can (x+3) or (x−3) go further? No. Finished
Stopping at 5x(x2 − 9) is the commonest answer to this question and it is only worth part marks.
Factorise fully 3x2 + 12x + 12
Pass 1 — every term divides by 3
3(x² + 4x + 4)
Pass 2 — two numbers multiplying to 4, adding to 4: 2 and 2
3(x + 2)(x + 2)
= 3(x + 2)²
When both brackets come out the same, write it as a square. It is tidier and it shows you noticed.
If any bracket you have written could be factorised again, you have not finished.
Solving quadratics by factorising
This is why factorising is worth learning properly. Once an expression is a product, you can solve it, because of one fact:
If two things multiply to give zero, at least one of them must be zero.
Nothing else in maths behaves like that. Two things multiplying to give 12 could be almost anything. Two things multiplying to give zero forces one of them to be zero, and that turns one hard equation into two easy ones.
- Get everything on one side, so the equation reads “something = 0”.
- Factorise that something.
- Set each bracket equal to zero and solve them separately.
Solve x2 + 7x + 12 = 0
Already equals zero. Factorise
(x + 3)(x + 4) = 0
So either the first bracket is 0, or the second is
x + 3 = 0 → x = −3
x + 4 = 0 → x = −4
x = −3 or x = −4
Note the signs flip. The bracket (x + 3) gives the solution x = −3, not +3. Reading the answers straight out of the brackets without changing the sign is one of the most frequent errors in GCSE and IB maths alike.
Solve 2x2 + 7x + 3 = 0
(2x + 1)(x + 3) = 0
2x + 1 = 0 → 2x = −1 → x = −½
x + 3 = 0 → x = −3
x = −½ or x = −3
When the bracket is (2x + 1) there is an extra step — divide by the 2 at the end. Fractional solutions are completely normal here and are not a sign you have gone wrong.
To solve x2 − 5x = 0, factorise to x(x − 5) = 0, giving x = 0 or x = 5. Dividing both sides by x instead leaves x = 5 and loses the solution x = 0 completely, because you divided by something that might itself be zero. Factorise, never divide.
Factorise, factorize, factoring: the same thing
Searching this topic turns up several spellings and it worries students unnecessarily. They all mean exactly the same operation.
| Word | Where it is used | Meaning |
|---|---|---|
| Factorise / factorisation | UK, Ireland, India, Australia, and the IB | Write as a product |
| Factorize / factorization | The same word with American spelling | Identical |
| Factoring / factor | United States and Canada | Identical. “Factor the expression” means factorise it |
Use whichever spelling your school uses; examiners accept both. IB papers and UK exam boards use factorise, so that is the spelling used throughout this page.
One word that genuinely is different: prime factorisation means breaking a number into prime numbers, such as 60 = 22 × 3 × 5. Same idea, applied to numbers rather than algebra. If a question mentions primes or a factor tree, it wants that and not this.
Practice questions
Eight questions covering all five types. The answers are hidden — do each one on paper, then tap to check. And remember you can mark yourself: expand your answer and see if you get the question back.
1. Factorise 10x + 15
HCF of 10 and 15 is 5. No letter in both terms
= 5(2x + 3)
Check: 5 × 2x = 10x, 5 × 3 = 15
2. Factorise fully 8x2 − 12x
HCF of 8 and 12 is 4. Both terms have x
= 4x(2x − 3)
2(4x2 − 6x) and 4(2x2 − 3x) are both true and neither is fully factorised.
3. Factorise x2 − 49
Two terms, both squares, a minus between them
√(x²) = x, √49 = 7
= (x + 7)(x − 7)
4. Factorise x2 − 5x + 6
Last sign plus, middle sign minus, so BOTH numbers negative
Multiply to 6, add to −5
−2 and −3
= (x − 2)(x − 3)
5. Factorise x2 + 3x − 10
Last sign minus, so ONE of each. Bigger one takes the +
Multiply to −10, add to 3
+5 and −2
= (x + 5)(x − 2)
6. Factorise 6x2 − x − 2
a × c = 6 × (−2) = −12
Multiply to −12, add to −1
3 and −4
Split the middle term
6x² + 3x − 4x − 2
3x(2x + 1) − 2(2x + 1)
= (3x − 2)(2x + 1)
The middle term is just −x, which means b = −1. A missing number in front of a letter always means 1.
7. Factorise fully 4x3 − 16x
Pass 1 — common factor 4x
4x(x² − 4)
Pass 2 — x² − 4 is square minus square
= 4x(x + 2)(x − 2)
Three factors in the final answer. Stopping at 4x(x2 − 4) loses the last mark.
8. Solve 6x2 − x − 2 = 0
Factorise it first (question 6)
(3x − 2)(2x + 1) = 0
3x − 2 = 0 → 3x = 2 → x = ⅔
2x + 1 = 0 → 2x = −1 → x = −½
x = ⅔ or x = −½
Where the marks get lost
- Not taking out the common factor first. It makes everything afterwards harder and it is usually where the “fully” mark lives.
- Taking out a factor instead of the highest one. 2(6x2 + 4x) is true and still wrong.
- Stopping before the brackets are finished. Check every bracket you wrote. Can it go further?
- Sign errors in the two numbers. Read the sign of c first, then the sign of b, before you start hunting.
- Trying to factorise a sum of two squares. x2 + 9 does not factorise at this level. Writing “does not factorise” is the right answer.
- Guessing on a > 1 quadratics. Use the splitting method. Guessing works until it does not, and then it wastes five minutes.
- Reading solutions out of brackets without flipping the sign. (x + 3) gives x = −3.
- Dividing both sides by x when solving. It silently throws away the solution x = 0.
- Not checking. Expanding your answer takes ten seconds and catches almost everything above.
Expand what you wrote. If you get back to the question exactly, your factorising is correct. If you do not, it is wrong — and you have found that out before the examiner did. There is no other topic in maths where self-marking is this reliable, so use it every time.
The whole thing on one card
Photograph this before the paper.
Factorising — quick reference
- Common factor comes out first, every single time. Highest common factor, not just any.
- a2 − b2 = (a + b)(a − b). Two terms, both squares, a minus between.
- a2 + b2 does not factorise. Saying so is the correct answer.
- Simple quadratic: two numbers that multiply to c and add to b.
- Hard quadratic: multiply a by c, find two numbers, split the middle term, factorise in pairs.
- Signs: c positive → same signs. c negative → one of each, bigger takes b’s sign.
- “Fully” means go again. Look inside every bracket you wrote.
- Solving: get “= 0”, factorise, set each bracket to zero. The signs flip.
- Never divide by x — you lose the solution x = 0.
- Check by expanding. Ten seconds, catches nearly everything.
Factorising sits underneath quadratics, algebraic fractions, curve sketching and most of the algebra in IB Maths, so it is worth being quick at rather than merely able to do. Our IB revision notes carry it through into quadratics and functions.
Frequently asked questions
What is factorisation in maths?
How do you factorise an expression?
What does “factorise fully” mean?
How do you factorise a quadratic?
How do you factorise when the coefficient of x² is greater than 1?
What is the difference of two squares?
Can you factorise x² + 25?
How do you factorise with four terms?
How do you solve a quadratic by factorising?
Is factorise the same as factorize?
What is the difference between factorising and expanding?
Why do we factorise?
What is prime factorisation?
How do I know which factorising method to use?
What if the two numbers do not exist?
Algebra slowing you down in the exam?
Factorising is one of those skills where being able to do it and being quick at it are different things, and only the second one helps under time pressure. Our tutors teach IB and A-level maths and work through your own past papers rather than a generic worksheet.
Book a free 30-minute meetingMore on this topic in our IB Maths pages and the IB revision notes. Working on a maths IA? See our maths IA topic ideas, or read about IB maths tutoring.