Factorising: how to factorise an expression fully

What does factorise mean?

To factorise means to write something as a multiplication of its factors. A factor is a number (or term) that divides into another exactly.

You can factorise numbers: 12 can be written as 3 × 4, or 2 × 6, or 2 × 2 × 3. Writing a number as a product of prime numbers, like that last one, is called prime factorisation, and the prime factors guide covers it. Every one of these comes from the times tables: spotting factors quickly is mostly knowing the tables well.

Factorising an algebraic expression works the same way. 6x + 15 can be written as 3 × (2x + 5), usually written 3(2x + 5). The 3 and the bracket are its factors.

Factorising is expanding brackets in reverse

Factorising undoes expanding. Expanding brackets takes 3(2x + 5) and multiplies it out to 6x + 15. Factorising takes 6x + 15 and puts the brackets back. That gives you a built-in check: expand your answer, and you should get back to the question.

How to factorise, step by step

To factorise an expression by taking out a common factor:

1. Find the highest common factor (HCF) of the numbers in every term.
2. Check the letters. If every term has the same letter, that's a common factor too.
3. Write the HCF outside a bracket.
4. Divide each term by the HCF and write what's left inside the bracket.
5. Check by expanding.

Worked example: factorise 8x + 12

1. The factors of 8 are 1, 2, 4 and 8. The factors of 12 are 1, 2, 3, 4, 6 and 12. The highest common factor is 4.
2. Only one term has an x, so x isn't a common factor.
3. Write 4 outside: 4( ).
4. 8x divided by 4 is 2x, and 12 ÷ 4 = 3. So the inside is 2x + 3.

The answer is 4(2x + 3). Check: 4 × 2x is 8x, and 4 × 3 = 12. That's the question again.

Worked example: factorise 10x − 4

The highest common factor of 10 and 4 is 2. 10x divided by 2 is 5x, and 4 ÷ 2 = 2, so 10x − 4 = 2(5x − 2). The minus sign stays inside the bracket, between the two terms.

How to factorise algebra with letters in common

When every term has the same letter, the letter comes outside the bracket as well. Remember that x² means x × x, so it has a factor of x.

• Factorise x squared + 5x. Both terms have an x. x² divided by x is x, and 5x divided by x is 5, so x² + 5x = x(x + 5).
• Factorise 6xy + 9x. The HCF of 6 and 9 is 3, and both terms have an x, so the common factor is 3x. 6xy divided by 3x is 2y, and 9x divided by 3x is 3. The answer is 3x(2y + 3).

How to factorise fully

To factorise fully, take out the highest common factor, not just any common factor. If the bracket still has a common factor inside, the expression isn't fully factorised.

Worked example: fully factorise 12t + 20

2 goes into both, so 12t + 20 could be written 2(6t + 10). But 6t and 10 still share a factor of 2, so that's not fully factorised. The highest common factor of 12 and 20 is 4:

12t + 20 = 4(3t + 5)

Now 3t and 5 have no common factor except 1, so it's done.

Worked example: fully factorise 3h squared + 12h

The HCF of 3 and 12 is 3, and both terms have an h, so take out 3h. 3h² divided by 3h is h, and 12h divided by 3h is 4. The answer is 3h(h + 4).

Taking out only the 3, as 3(h² + 4h), or only the h, as h(3h + 12), leaves a common factor inside the bracket, so neither is fully factorised.

Do you factorise an equation or an expression?

You factorise an expression, not an equation. An expression like 4x + 8 has no equals sign, and factorising it gives 4(x + 2): the same thing written another way. An equation, like 4x + 8 = 20, is something you solve to find x, and the solving equations guide shows how.

The two do meet at GCSE. The key stage 4 curriculum includes factorising quadratic expressions such as x² + 5x + 6 into two brackets, (x + 2)(x + 3), and using that to solve quadratic equations. That's a separate topic, factorising quadratics, and it builds directly on the common factors on this page. In the DfE's GCSE content, quadratics that start with a number in front of the x², such as 2x² + 7x + 3, are marked for the more highly attaining students only.

Common mistakes when factorising

• Not factorising fully. 2(6t + 10) is a correct factorisation of 12t + 20, but not a full one. Check the bracket for a common factor.
• Leaving out a term. When a term is the same as the common factor, a 1 is left behind: 5x + 5 = 5(x + 1), not 5(x).
• Forgetting the letter. In x² + 7x, the x is a common factor, so the answer is x(x + 7).
• Losing the sign. 15y − 10 is 5(3y − 2). The minus goes inside the bracket.
• Not checking. Expanding the answer takes seconds and catches nearly every slip.

How to help at home

• Practise factors of numbers. "What's the biggest number that goes into 18 and 24?" Quick answers to questions like this make factorising much easier. The times table grid is handy for spotting common factors.
• Ask "what's left?" When your child takes out a factor, ask what each term divides to. Saying it out loud slows down the guessing.
• Expand to check. Make it a habit: factorise, then expand to see the question come back.
• Look for more. Ask "can anything else come out?" after every answer, until the answer is no.

Factorising questions to try

1. Factorise 5x + 10.
2. Factorise 12x − 8.
3. Factorise 7y + 21.
4. Factorise x² − 6x.
5. Fully factorise 18a + 24.
6. Fully factorise 4p² + 16p.

Answers

1. 5(x + 2)
2. 4(3x − 2)
3. 7(y + 3)
4. x(x − 6)
5. 6(3a + 4)
6. 4p(p + 4)

For the rest of what children learn in Years 7 to 9, see the KS3 maths guide.