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IGCSE Maths · Edexcel 4XMA1 · Unit 1 Higher · Algebraic manipulation

Expanding Triple Brackets

Expanding three brackets is a standard 3-mark question on the Higher paper. It also feeds into proofs, volumes and cubic graphs. The method is always the same: expand two brackets, then multiply the result by the third.

1The method: two brackets, then the third

You already know how to expand two brackets. To expand three, do it in two stages:

  1. Expand any two of the brackets and simplify (you should get a quadratic).
  2. Multiply every term of that quadratic by every term of the third bracket.
  3. Collect like terms. Your answer should have an x³ term, an x² term, an x term and a number.
Worked example
Expand (x + 2)(x + 3)(x + 4).
  1. First two brackets: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.
  2. Multiply by (x + 4): x²(x + 4) + 5x(x + 4) + 6(x + 4) = x³ + 4x² + 5x² + 20x + 6x + 24.
  3. Collect like terms: x³ + 9x² + 26x + 24.
x³ + 9x² + 26x + 24
CheckA quadratic (3 terms) times a bracket (2 terms) gives 3 × 2 = 6 terms before you simplify. If you have fewer than 6, you have missed one.

2Using a grid for the second stage

A grid keeps the six products organised. Put the quadratic along the top and the third bracket down the side:

×x²+5x+6
xx³5x²6x
+44x²20x24

Now add the diagonals of like terms: x³ + (5x² + 4x²) + (6x + 20x) + 24 = x³ + 9x² + 26x + 24.

3Brackets with negative terms

Take care with signs. Write each product with its sign and simplify at the end.

Worked example
Expand and simplify (x − 1)(x + 2)(x − 3).
  1. (x − 1)(x + 2) = x² + 2x − x − 2 = x² + x − 2.
  2. (x² + x − 2)(x − 3) = x³ − 3x² + x² − 3x − 2x + 6.
  3. Collect: x³ − 2x² − 5x + 6.
x³ − 2x² − 5x + 6
Exam tipQuick check: substitute x = 1 into the question and into your answer. (0)(3)(−2) = 0 and 1 − 2 − 5 + 6 = 0 ✓. Checking with x = 2 as well catches almost every slip.

4Coefficients other than 1

Worked example
Expand and simplify (2x + 1)(x − 3)(x + 2).
  1. (2x + 1)(x − 3) = 2x² − 6x + x − 3 = 2x² − 5x − 3.
  2. (2x² − 5x − 3)(x + 2) = 2x³ + 4x² − 5x² − 10x − 3x − 6.
  3. Collect: 2x³ − x² − 13x − 6.
2x³ − x² − 13x − 6

The x³ coefficient is the product of the three x-coefficients (2 × 1 × 1 = 2), and the constant is the product of the three numbers (1 × −3 × 2 = −6). Use these as a quick check.

5Cubing a bracket

(a + b)³ means (a + b)(a + b)(a + b). Expand (a + b)² first, then multiply by (a + b) once more.

Worked example
Expand (x + 3)³.
  1. (x + 3)² = x² + 6x + 9.
  2. (x² + 6x + 9)(x + 3) = x³ + 3x² + 6x² + 18x + 9x + 27.
  3. Collect: x³ + 9x² + 27x + 27.
x³ + 9x² + 27x + 27
Worked example
Expand (2x − 1)³.
  1. (2x − 1)² = 4x² − 4x + 1.
  2. (4x² − 4x + 1)(2x − 1) = 8x³ − 4x² − 8x² + 4x + 2x − 1.
  3. Collect: 8x³ − 12x² + 6x − 1.
8x³ − 12x² + 6x − 1
Classic error
  • (x + 3)³ is not x³ + 27. You must expand fully.

6Finding unknown coefficients

Some questions give the expanded answer with letters for unknown coefficients. Expand your side fully, then compare coefficients of each power of x.

Worked example
(x + a)(x + 2)(x − 1) ≡ x³ + bx² + cx − 12. Find a, b and c.
  1. (x + 2)(x − 1) = x² + x − 2.
  2. (x + a)(x² + x − 2) = x³ + x² − 2x + ax² + ax − 2a = x³ + (1 + a)x² + (a − 2)x − 2a.
  3. Constant terms: −2a = −12, so a = 6.
  4. x² terms: b = 1 + a = 7. x terms: c = a − 2 = 4.
a = 6, b = 7, c = 4
Key ideaThe ≡ symbol means “identically equal”: true for every value of x. That is why matching coefficients works.
Common mistakes
  • Trying to multiply all three brackets at once and missing terms. Always go two, then three.
  • Sign errors when multiplying negatives: (−2)(−3) = +6.
  • Writing (x + 3)³ = x³ + 27, or (2x)³ = 2x³ instead of 8x³.
  • Not collecting like terms at the end, which loses the final accuracy mark.

★Key Vocabulary Recap

Expand
Multiply out brackets to remove them.
Simplify
Collect like terms so each power of x appears once.
Coefficient
The number in front of a term, e.g. −13 in −13x.
Cubic expression
An expression whose highest power of x is 3.
Identity (≡)
An equation that is true for all values of the variable.
Specification reference (Pearson Edexcel 4XMA1, Unit 1 Higher)
  • Expand the product of two or more linear expressions.
  • Manipulate algebraic expressions, including collecting like terms and comparing coefficients.
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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