Expanding Triple Brackets –

Expanding triple brackets is a Higher tier GCSE Maths skill that builds directly on expanding double brackets. You produce a cubic expression by expanding two brackets first, then multiplying the result by the third bracket.

What Is Expanding Triple Brackets?

When three linear brackets are multiplied together, such as (x + 2)(x + 3)(x - 1), the result is a cubic expression — one that contains an x³ term. You cannot use FOIL alone because there are three factors, so the process is split into two stages.

Stage 1: Pick any two of the three brackets and expand them using the standard double-bracket method (FOIL or grid). This gives a quadratic expression.

Stage 2: Multiply every term of that quadratic by every term of the remaining bracket, then collect like terms. The final answer will typically have four terms: an x³ term, an x² term, an x term, and a constant.

Getting comfortable with this process is important because it links to sketching cubic graphs and solving cubic equations at the Higher tier.

Key Formulas

Step-by-Step Method

1. Choose two of the three brackets (usually the first two) and expand them to get a quadratic.
2. Write the quadratic expression clearly, collecting like terms.
3. Multiply each term of the quadratic by the first term of the third bracket.
4. Multiply each term of the quadratic by the second term of the third bracket.
5. Combine all terms and collect like terms to write the final cubic expression.

Worked Example 1 — Foundation Level

Question: This is a Higher only topic, but here is a straightforward example. Expand (x + 1)(x + 2)(x + 3).

Working:

Step 1 — Expand the first two brackets: (x + 1)(x + 2) = x² + 2x + x + 2 = x² + 3x + 2.

Step 2 — Multiply by the third bracket: (x² + 3x + 2)(x + 3).

Step 3 — Multiply each term by x: x³ + 3x² + 2x.

Step 4 — Multiply each term by 3: 3x² + 9x + 6.

Step 5 — Collect like terms: x³ + 3x² + 3x² + 2x + 9x + 6 = x³ + 6x² + 11x + 6.

Answer: x³ + 6x² + 11x + 6

Worked Example 2 — Higher Level

Question: Expand and simplify (x - 2)(x + 4)(x - 3).

Working:

Step 1 — Expand the first two brackets: (x - 2)(x + 4) = x² + 4x - 2x - 8 = x² + 2x - 8.

Step 2 — Multiply by (x - 3): (x² + 2x - 8)(x - 3).

Step 3 — Multiply each term by x: x³ + 2x² - 8x.

Step 4 — Multiply each term by -3: -3x² - 6x + 24.

Step 5 — Collect like terms: x³ + 2x² - 3x² - 8x - 6x + 24 = x³ - x² - 14x + 24.

Answer: x³ - x² - 14x + 24

Worked Example 3 — Exam Style

Question: Show that (2x + 1)(x - 1)(x + 3) = 2x³ + 5x² - 4x - 3. (4 marks)

Working:

Step 1 — Expand (2x + 1)(x - 1): 2x² - 2x + x - 1 = 2x² - x - 1.

Step 2 — Multiply by (x + 3): (2x² - x - 1)(x + 3).

Step 3 — Multiply each term by x: 2x³ - x² - x.

Step 4 — Multiply each term by 3: 6x² - 3x - 3.

Step 5 — Collect like terms: 2x³ - x² + 6x² - x - 3x - 3 = 2x³ + 5x² - 4x - 3.

This matches the right-hand side. QED.

Answer: Shown: the expansion gives 2x³ + 5x² - 4x - 3.

Common Mistakes

• Trying to expand all three brackets at once. Always expand two brackets first to get a quadratic, then multiply by the third. Skipping this step leads to missing terms.
• Sign errors in the second multiplication. When the third bracket contains a negative term, every product with that term changes sign. Track negatives carefully.
• Forgetting to collect all like terms. After the second expansion you will have six terms. There are usually two x² terms and two x terms to combine.

Exam Tips

• It does not matter which two brackets you expand first — pick the pair that looks simplest.
• Use a grid or table layout to organise the second multiplication if you find it hard to keep track.
• Verify your answer by substituting x = 1 into both the original brackets and the expanded form.

Practice Questions

Q1 (Higher): Expand and simplify (x + 2)(x + 5)(x + 1).

Q2 (Higher): Expand and simplify (x - 1)(x + 3)(x - 4).

Q3 (Higher): Expand and simplify (2x - 1)(x + 2)(x - 3).

Practise expanding triple brackets questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Expanding Double Brackets
• Expanding Brackets
• Factorising Quadratics

Summary

• To expand triple brackets, first expand two of the brackets to get a quadratic expression.
• Then multiply every term of the quadratic by every term of the remaining bracket.
• The result is a cubic expression with up to four terms.
• Collect like terms carefully — there will be multiple x² and x terms to combine.
• Check your answer by substituting a simple value like x = 1 into both forms.