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Free GCSE maths resource

GCSE simplifying, expanding and factorising practice

See how like terms, brackets and common factors connect. Learn to simplify expressions and check a factorisation by expanding it again.

Foundation and Higher · Useful for students strengthening core skills before mixed GCSE practice.

Core pathway. Core GCSE methods are useful across Foundation and Higher. Start with the core examples and questions.

Higher extension. More demanding selections are marked Higher extension: practice question 8. Use these after the core work, following your teacher’s guidance.

3 worked examples and 8 original practice questions · Allow 25–40 minutes · Read online or print. Higher extensions are labelled. No account or email needed.

Learn the methodTry the questions

Printing includes the method, examples and space for working, followed by a separate answer section.

A useful approach

  1. Collect like terms with matching letters and powers, keeping their signs.
  2. Expand by multiplying every term; factorise by taking out a common factor.
  3. For quadratic factors, check the sum and product of the constants. Expand your answer to check it.

A mistake to watch for

An expression and an equation are different tasks. Simplify an expression; solve an equation for the unknown. A squared term cannot be combined with an ordinary linear term.

Learn each method, then practise it

Read each line of working and explain why it follows from the previous line.

Collecting like terms

Only combine terms with the same letter part and power. Carry each sign with its term.

Worked example 1

Simplifying Algebra · Core GCSE skill · No calculator · 1 marks

Simplify  7c − 2c + 4c

  1. The coefficient is the number multiplying a letter. Add the coefficients of matching letter terms, just as 4 lots of an object plus 5 lots make 9 lots.
  2. All the terms are in c, so combine the coefficients: 7 - 2 + 4 = 9.
  3. 7c - 2c + 4c = 9c.
Answer: 9c
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Now practise this method: Question 1 · Question 2

Expanding and taking out a common factor

To expand, multiply every term in a bracket. To factorise, divide every term by their common factor and place what remains inside brackets.

Worked example 2

Expanding and Factorising · Core GCSE skill · Calculator allowed · 2 marks

  1. Expand 4(2x − 5)
  2. Factorise 6y + 15
  1. a) 4 times 2x = 8x and 4 times -5 = -20, so 8x - 20
  2. b) The highest common factor of 6y and 15 is 3
  3. 6y + 15 = 3(2y + 5)
  4. Factorising reverses expanding: divide every term by the common factor to find what belongs inside the bracket, then expand again to check.
Answer: (a) 8x - 20   (b) 3(2y + 5)
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Now practise this method: Question 3 · Question 4

Two brackets and quadratic factors

Multiply each term in the first bracket by each term in the second. Reverse that process using two numbers with the required sum and product.

Worked example 3

Expanding and Factorising Quadratics · Core GCSE skill · Calculator allowed · 2 marks

Factorise  x2 + 9x + 14

  1. Expanding (x + a)(x + b) gives x² + (a + b)x + ab. Therefore the two numbers must add to the x coefficient and multiply to the constant.
  2. Find two numbers that multiply to 14 and add to 9: they are 2 and 7
  3. x2 + 9x + 14 = (x + 2)(x + 7)
Answer: (x + 2)(x + 7)
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Now practise this method: Question 5 · Question 6 · Question 7 · Question 8

Your practice questions

Write your working on paper. Marks indicate how much working to show; these questions are self-marked and do not change saved practice results. Use squared paper for drawing questions.

Question 8

Expanding and Factorising Quadratics · Higher extension · No calculator · 3 marks

A rectangular card measures (x + 5) cm by (x + 3) cm.

A square of side x cm is cut from the card.

Show that the area of card remaining is (8x + 15) cm2.

(x + 5) cm(x + 3) cmx cmx cmCut outDiagram NOT accurately drawn
Diagram for Expanding and Factorising Quadratics. The question states the required measurements and relationships.
Read a text description of this diagram

Given diagram. A rectangular card measures ( x + 5) cm by ( x + 3) cm. A square of side x cm is cut from the card. Show that the area of card remaining is (8 x + 15) cm to the power 2 . Use the labelled information; do not estimate measurements from the sketch.

Check answer and working for question 8

Report a problem with this question

Worked answers: Simplifying, expanding and factorising

Compare the reasoning as well as the final answer. Another correct method is valid. If a step is unclear, revisit an example before trying a similar question.

Answer 1

Show answer and working for question 1
  1. There are four lots of a added together.
  2. a + a + a + a = 4a.
Answer: 4a
Back to question 1

Answer 2

Show answer and working for question 2
  1. Like terms have exactly the same letters and powers. Combine x terms with x terms and y terms with y terms; an x and a y need not have the same value.
  2. Collect the x terms: 6x - 2x = 4x.
  3. Collect the y terms: 3y + 5y = 8y.
  4. The simplified expression is 4x + 8y.
Answer: 4x + 8y
Back to question 2

Answer 3

Show answer and working for question 3
  1. a) The highest common factor of 5x and 20 is 5, so 5x + 20 = 5(x + 4)
  2. b) Both terms contain x, so x2 - 9x = x(x - 9)
Answer: (a) 5(x + 4)   (b) x(x - 9)
Back to question 3

Answer 4

Show answer and working for question 4
  1. a) Every term inside the bracket must be multiplied by 3
  2. 3(2x + 4) = 6x + 12, not 6x + 4
  3. b) The highest common factor of 12x2 and 18x is 6x
  4. 12x2 + 18x = 6x(2x + 3)
Answer: (a) He only multiplied the first term by 3; 3 times 4 = 12, so the answer should be 6x + 12   (b) 6x(2x + 3)
Back to question 4

Answer 5

Show answer and working for question 5
  1. (x + 4)(x + 6) = x2 + 6x + 4x + 24
  2. Collect like terms: x2 + 10x + 24
Answer: x2 + 10x + 24
Back to question 5

Answer 6

Show answer and working for question 6
  1. (x - 3)(x + 8) = x2 + 8x - 3x - 24
  2. Collect like terms: x2 + 5x - 24
Answer: x2 + 5x - 24
Back to question 6

Answer 7

Show answer and working for question 7
  1. a) (x + 9)(x - 2) = x2 - 2x + 9x - 18 = x2 + 7x - 18
  2. b) Find two numbers that multiply to -28 and add to -3: they are -7 and 4
  3. b) x2 - 3x - 28 = (x - 7)(x + 4)
Answer: (a) x2 + 7x - 18   (b) (x - 7)(x + 4)
Back to question 7

Answer 8

Show answer and working for question 8
  1. Area of rectangle = (x + 5)(x + 3) = x2 + 3x + 5x + 15 = x2 + 8x + 15
  2. Area of square removed = x2
  3. Remaining area = x2 + 8x + 15 - x2 = 8x + 15
Answer: (x + 5)(x + 3) - x2 = x2 + 8x + 15 - x2 = 8x + 15
Back to question 8

Practise a specific skill

Workouts provide interactive practice; choose the tier your school has recommended.

Use this in class

Compare expanding with factorising as inverse processes. Ask students to check each factorisation by expanding.

Build a custom classroom worksheet or print this page. The teacher builder can display questions on a projector and reveal answers separately.

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