Factorising into Single Brackets
Factorising into a single bracket is the reverse process of expanding: writing an expression as a common factor multiplied by a bracket, for example 6x + 15 = 3(2x + 5). Full factorisation uses the highest common factor (HCF) of every term in the expression, which may include both a shared number and a shared letter.
Method
- Find the highest common factor (HCF) of the numerical coefficients of every term in the expression.
- Check whether every term also shares a common letter, or power of a letter; if so, include it in the common factor.
- Combine the numerical HCF and any common letter to find the overall HCF of the expression.
- Divide each term in the expression by this HCF to find what goes inside the bracket.
- Write the HCF outside the bracket and the results of dividing inside the bracket.
- Check the answer by expanding the bracket back out - it should exactly match the original expression.
Worked example
Factorise fully 12x^2 + 8x.
- Find the HCF of the coefficients 12 and 8, which is 4.
- Check the letters: both terms contain at least one x, so x is a common factor too.
- The overall HCF of the expression is 4x.
- Divide each term by 4x: 12x^2 / 4x = 3x, and 8x / 4x = 2.
- Write the answer: 4x(3x + 2).
- Check by expanding: 4x x 3x = 12x^2, and 4x x 2 = 8x, which matches the original expression.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Factorise fully 5x + 20.Show answer
Answer: 5(x + 4)
Q2Factorise fully 6x - 9.Show answer
Answer: 3(2x - 3)
Q3Factorise fully 4x + 4y.Show answer
Answer: 4(x + y)
Q4Factorise fully 10x^2 + 15x.Show answer
Answer: 5x(2x + 3)
Q5Factorise fully 9x^2 - 6x.Show answer
Answer: 3x(3x - 2)
Q6Factorise fully 7a + 14ab.Show answer
Answer: 7a(1 + 2b)
Q7Factorise fully 8xy + 12x.Show answer
Answer: 4x(2y + 3)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Factorise fully 18x^2 + 12x.
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A student factorises 20x + 15 as 5(4x + 3). Show that this is a correct factorisation, and explain why 5(4x + 3) is the FULL factorisation rather than using a smaller common factor.
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Free printable worksheet
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This topic is chapter 24 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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