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2.6 Factorising

GCSE Maths Foundation · Edexcel · Chapter 2: Expressions

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: taking out common factors to factorise expressions into a single bracket. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Understand that factorising is the reverse of expanding
  • Factorise expressions by taking out the highest common factor of the numbers
  • Factorise expressions with a common letter, e.g. x² + 6x and 8y² − 12y
  • Check a factorisation by expanding, and use factorising to solve problems

Key words

  • Factorise: Write an expression as a product, using brackets: 5x + 20 = 5(x + 4).
  • Common factor: A number or letter that divides into every term.
  • Fully factorise: Take out the highest common factor, so nothing more can come out of the bracket.

Factorising is the reverse of expanding

To factorise, find a factor of every term, write it outside the bracket, then divide each term by it to find what goes inside.

Worked example: a number factor

Factorise 5x + 20
5 divides 5x and 20. 5x ÷ 5 = x, 20 ÷ 5 = 4
Check: 5(x + 4) = 5x + 20 ✓

Answer: 5(x + 4)

Taking out the highest common factor

Always use the highest common factor (HCF). If the terms in the bracket still share a factor, you have not finished.

Worked example: factorise fully

Factorise fully 12a − 18
Common factors of 12 and 18: 2, 3, 6. The HCF is 6.
12a ÷ 6 = 2a, 18 ÷ 6 = 3

Answer: 6(2a − 3)

Fully?: 2(6a − 9) is correct in value but not fully factorised, because 6 and 9 still share a factor of 3.

Factorising with letters

A letter can be a common factor too. Look for the HCF of the numbers and the letters every term shares.

Worked example: letter factors

(a) x² + 6x = x(x + 6)
(b) 8y² − 12y: HCF of 8 and 12 is 4, both terms contain y, so take out 4y: 4y(2y − 3)
(c) 6ab + 9a = 3a(2b + 3)

Answer: x(x + 6), 4y(2y − 3), 3a(2b + 3)

Checking and using factorising

Always check by expanding your answer. Factorising also helps with areas and with showing facts about numbers.

Worked example: even numbers

Show that 4n + 6 is always even when n is a whole number.
4n + 6 = 2(2n + 3). It is 2 times a whole number, so it is even.

Answer: 2(2n + 3)

Worked example: quicker arithmetic

Work out 17 × 23 + 17 × 77
= 17(23 + 77) = 17 × 100 = 1700

Answer: 1700

Common mistakes

  • Not factorising fully: 12a + 18 = 2(6a + 9) is not finished. The answer is 6(2a + 3).
  • Forgetting the 1: x² + x = x(x + 1), not x(x).
  • Dividing only the first term: 5x + 20 is not 5(x + 20).
  • Missing a common letter: 8y² − 12y = 4y(2y − 3), not 4(2y² − 3y).

Revision questions

Try these, then check your answers below each one.

Question 1. Factorise 6x + 18. (Choose from: 6(x + 3), 3(2x + 18), 6(x + 18), 6x(1 + 3).)

Answer: 6(x + 3)

Working: The HCF is 6. 6x ÷ 6 = x, 18 ÷ 6 = 3, so 6(x + 3).

Question 2. Factorise fully 10y - 25. (Choose from: 5(2y - 5), 5(2y - 25), 10(y - 25), 2(5y - 25).)

Answer: 5(2y - 5)

Working: 10y ÷ 5 = 2y, 25 ÷ 5 = 5, so 5(2y - 5).

Question 3. Factorise x² + 4x. (Choose from: x(x + 4), x(x + 4x), 4x(x + 1), x²(1 + 4).)

Answer: x(x + 4)

Working: x² ÷ x = x, 4x ÷ x = 4, so x(x + 4).

Related lessons

Continue in the full course

This lesson is section 2.6 (EGCMF0206) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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