Topics Grade boundaries Exam dates Yumio GCSE Maths Higher Grade 6 Expanding triple brackets
Expanding triple brackets – GCSE Maths Higher · Grade 6 · Non-calculator
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On this page Objectives Key points Examples Questions Mistakes Tips Worksheet Expand a product of three brackets by multiplying two first, then multiplying the result by the third. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Three brackets: expand two of them first, then multiply the quadratic by the remaining linear. ( x + 1 ) ( x + 2 ) ( x + 3 ) = ( x 2 + 3 x + 2 ) ( x + 3 ) = x 3 + 6 x 2 + 11 x + 6 x 2 x
Check by putting x = 1 x 2
Multiply two brackets first, then the third. A difference of two squares can shrink the first pair. ( x + 2 ) 3 x 3 + 8 ( x + 2 ) 2 ( x + 2 )
Objectives Expand three linear brackets into a cubic. Expand a cubed bracket. Read the coefficient of x^2 without writing every term. Key points Expand two linear brackets into a quadratic confidently. Multiply a quadratic by a linear bracket, keeping every one of the six terms. Collect like terms and write the answer in descending powers of x Recognise a difference of two squares such as ( x − 1 ) ( x + 1 ) = x 2 − 1 Worked examples Example 1 Expand (x+1)(x+2)(x+3).
2 marks
Expand the first two brackets: (x+1)(x+2) = x^2 + 3x + 2. Multiply the quadratic by the third bracket: (x^2 + 3x + 2)(x+3). Expand and collect terms: x^3 + 3x^2 + 3x^2 + 9x + 2x + 6 = x^3 + 6x^2 + 11x + 6. Answer: x^3+6x^2+11x+6
Common slips Slip: Multiply only the constants: 1 × 2 × 3 = 6. All terms in each bracket must be multiplied. Slip: Stop at x^2 + 3x + 2. The third bracket (x+3) must still be multiplied. Slip: Leave unsimplified terms. Combine all like terms for the final cubic. Example 2 Expand (x+1)^3.
2 marks
Write as three repeated linear factors: (x+1)(x+1)(x+1). Expand the first pair: (x+1)^2 = x^2 + 2x + 1. Multiply by (x+1): x^3 + 3x^2 + 3x + 1. Answer: x^3+3x^2+3x+1
Common slips Slip: Cube each term separately: x^3 + 1. Cubing a bracket produces cross terms. Slip: Write x^2 + 2x + 1. That is (x+1)^2. A cube needs one more bracket multiplied in. Slip: Drop the middle terms. Binomial coefficients for a cube are 1, 3, 3, 1. Example 3 Find the coefficient of x^2 in (x+1)(x+2)(x+3).
3 marks
Recognise that the x^2 term is formed by choosing x from two brackets and a constant from the third. Sum the three constants: 1 + 2 + 3 = 6. The coefficient of x^2 is 6. Answer: 6
Common slips Slip: Multiply all three constants together: 1 × 2 × 3 = 6. Multiplying all constants gives the constant term, not the x^2 coefficient. Slip: Multiply 1 and 2. Each bracket contributes a constant to one of the x^2 terms. Slip: State 6x^2. The question asks for the numerical coefficient. Questions and answers GCSE Maths expanding triple brackets questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy 2 marks Expand (x+1)(x+2)(x+4).
Show answer Answer: x^3+7x^2+14x+8
First pair: (x+1)(x+2) = x^2 + 3x + 2. Multiply by (x+4): x^3 + 7x^2 + 14x + 8. M1 — First pair: (x+1)(x+2) = x^2 + 3x + 2. A1 — Multiply by (x+4): x^3 + 7x^2 + 14x + 8. Question 2 Easy 2 marks Expand (x+2)(x+3)(x+4).
Show answer Answer: x^3+9x^2+26x+24
First pair: (x+2)(x+3) = x^2 + 5x + 6. Multiply by (x+4): x^3 + 9x^2 + 26x + 24. M1 — First pair: (x+2)(x+3) = x^2 + 5x + 6. A1 — Multiply by (x+4): x^3 + 9x^2 + 26x + 24. Question 3 Medium 2 marks Expand (x+2)^3.
Show answer Answer: x^3+6x^2+12x+8
Expand (x+2)^2 = x^2 + 4x + 4. Multiply by (x+2): x^3 + 6x^2 + 12x + 8. M1 — Expand (x+2)^2 = x^2 + 4x + 4. A1 — Multiply by (x+2): x^3 + 6x^2 + 12x + 8. Question 4 Medium 2 marks Expand (x+3)^3.
Show answer Answer: x^3+9x^2+27x+27
Expand (x+3)^2 = x^2 + 6x + 9. Multiply by (x+3): x^3 + 9x^2 + 27x + 27. M1 — Expand (x+3)^2 = x^2 + 6x + 9. A1 — Multiply by (x+3): x^3 + 9x^2 + 27x + 27. Question 5 Hard 2 marks Find the coefficient of x^2 in (x+1)(x+2)(x+4).
Show answer Answer: 7
Sum the constants from each factor. 1 + 2 + 4 = 7. M1 — Sum the constants from each factor. A1 — 1 + 2 + 4 = 7. Question 6 Hard 2 marks Find the coefficient of x^2 in (x+2)(x+3)(x+4).
Show answer Answer: 9
Sum the constants from each factor. 2 + 3 + 4 = 9. M1 — Sum the constants from each factor. A1 — 2 + 3 + 4 = 9. Question 7 Exam-style 4 marks Expand (x+1)(x+3)(x+4).
Show answer Answer: x^3+8x^2+19x+12
Expand the first pair: (x+1)(x+3) = x^2 + 4x + 3. Multiply by (x+4): x(x^2 + 4x + 3) + 4(x^2 + 4x + 3). Expand: x^3 + 4x^2 + 3x + 4x^2 + 16x + 12. Collect like terms: x^3 + 8x^2 + 19x + 12. M1 — Expand first pair to x^2 + 4x + 3 M1 — Multiply quadratic by (x+4) M1 — Collect x^2 terms 8x^2 and x terms 19x A1 — Final expansion x^3 + 8x^2 + 19x + 12 Common mistakes Most lost marks on this topic are one slip.
Stopping after two of the three brackets
Multiply by the third bracket
Writing x^3 plus the constant and nothing else for a cube
Include both middle terms
Multiplying the three numbers when the question asks for the x^2 coefficient
Add those three numbers
Spot the slip Ada expands (x+1)(x+2)(x+3) and stops after two brackets. The cubic should be which polynomial?
x^3+6x^2+11x+6 x^2+3x+2 x^3+6x+6 Show the slip x^3+6x^2+11x+6 The three constants add to the x^2 coefficient: 1+2+3 = 6, and the constant term is 1×2×3.
Exam tips Be ready to do this without a calculator. Paper 1 is the non-calculator paper. If the question says “show that”, the result is given. The marks are for the steps that reach it. The method is the same on AQA-style and Edexcel-style papers. Write the quadratic from the first pair on its own line; it is usually worth a mark on its own. Use the quick checks: the constant should be the three constants multiplied (1 × 2 × 4 = 8 x 3 x Printable worksheet The same questions are in a PDF you can print. Answers start on a later page.
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Questions students ask Is Expanding triple brackets Foundation or Higher? Expanding triple brackets is Higher only, aimed at grade 6.
What is a common slip on Expanding triple brackets? Stopping after two of the three brackets. Multiply by the third bracket.
Where can I get Expanding triple brackets questions and answers? The questions on this page are free. Open Show answer under each one, or download the worksheet PDF. Start free to practise this topic in the app.
Related topics More on Expanding triple brackets Previous topic Exam technique: “show that” questions Next topic Parallel & perpendicular lines AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.
Last updated 9 November 2026.