How to Expand Triple Brackets Step by Step

Expanding triple brackets is a GCSE and IGCSE algebra skill where three factors are multiplied to form a simplified cubic expression. The most reliable approach is to expand two brackets first (using FOIL or the grid method), then distribute the result across the third bracket, carefully collecting like terms. This page walks through the process step by step with clear worked examples, highlights common sign mistakes, and includes practice questions designed to build exam confidence, speed, and accuracy.

Prior Knowledge Be confident expanding a single bracket and the product of two brackets first. If you need a refresher, work through Expanding Double Brackets before starting here.
⇩ Jump to Practice Questions ⇩
What it means
Expanding triple brackets means multiplying three factors out and writing the result as a single simplified cubic expression.
The method
Expand two brackets first, then multiply that quadratic by the third bracket, and collect like terms.
The shape of the answer
\(x^3 + \ldots\)
Three linear brackets always give a cubic: the highest power is \(x^3\). Write your answer in descending powers.

How to Expand Three Brackets

  1. Pick any two brackets and expand them first, using the grid (area) method or FOIL.
  2. Write the quadratic result neatly, for example \(x^2 + bx + c\).
  3. Multiply every term of that quadratic by every term of the third bracket.
  4. Collect like terms and present the final answer in descending powers of \(x\).

For the first two brackets, the FOIL method gives four products: First, Outer, Inner, Last. The loops show which terms multiply together.

FOIL Method

( x + 3 ) ( x + 2 ) First Outer Inner Last
First Outer Inner Last

The four products of \((x+3)(x+2) = x^2 + 5x + 6\).

Grid (Area) Method

x +3 x +2 x² 3x 2x 6

The same four products, shown as an area (box) model.

⚠️ Watch the Signs

Most marks here are lost on signs. Always carry the sign with its term. With three brackets there are more places to slip, so check every sign before collecting like terms.

(+) × (+) = (+)
(−) × (−) = (+)
(+) × (−) = (−)
(−) × (+) = (−)
General result
\((x+a)(x+b)(x+c) = x^3 + (a+b+c)\,x^2 + (ab+ac+bc)\,x + abc\)

Worked Examples

💡 Example 1: FOIL two brackets

Expand \((x+3)(x+5)\).

First: \(x \times x = x^2\)
Outer: \(x \times 5 = 5x\)
Inner: \(3 \times x = 3x\)
Last: \(3 \times 5 = 15\)
Uncollected: \(x^2 + 5x + 3x + 15\)
Answer: \(x^2 + 8x + 15\)
What's happening?

FOIL gives the four products from a pair of brackets. This quadratic is exactly what you carry into the triple-bracket questions, so get it tidy first.

💡 Example 2: Three brackets

Expand \((x+1)(x+4)(x-2)\).

Step 1: FOIL the first two brackets
First: \(x \times x = x^2\)
Outer: \(x \times 4 = 4x\)
Inner: \(1 \times x = x\)
Last: \(1 \times 4 = 4\)
Gives: \(x^2 + 5x + 4\)
Step 2: Multiply by the third bracket \((x-2)\)
\(x^2(x-2) = x^3 - 2x^2\)
\(5x(x-2) = 5x^2 - 10x\)
\(4(x-2) = 4x - 8\)
Answer: \(x^3 + 3x^2 - 6x - 8\)
What's happening?

FOIL handles the first two brackets; then each term of that quadratic is distributed across the third bracket. Collect like terms: \(-2x^2 + 5x^2 = 3x^2\) and \(-10x + 4x = -6x\).

💡 Example 3: A bracket with a coefficient

Expand \((2x-1)(x+3)(x-2)\).

Step 1: FOIL the first two brackets
First: \(2x \times x = 2x^2\)
Outer: \(2x \times 3 = 6x\)
Inner: \((-1) \times x = -x\)
Last: \((-1) \times 3 = -3\)
Gives: \(2x^2 + 5x - 3\)
Step 2: Multiply by \((x-2)\)
\(2x^2(x-2) = 2x^3 - 4x^2\)
\(5x(x-2) = 5x^2 - 10x\)
\((-3)(x-2) = -3x + 6\)
Answer: \(2x^3 + x^2 - 13x + 6\)
What's happening?

Watch the signs from the \(-1\), \(-2\) and \(-3\), and let the leading \(2\) carry through to \(2x^3\). Collect: \(-4x^2 + 5x^2 = x^2\) and \(-10x - 3x = -13x\).

🔑 Key Points

  • Expand two brackets first, then multiply the quadratic by the third bracket.
  • Multiply every term by every term: a quadratic times a linear bracket gives six products before collecting.
  • Three linear brackets give a cubic: the highest power is \(x^3\).
  • Collect like terms and write the answer in descending powers of \(x\).
  • The order you pick the brackets in does not change the final answer.

⚠️ Common Pitfalls

  • Sign slips with negative terms. Write the sign first, the number second, and carry it carefully.
  • Forgetting to multiply one of the terms, so a term is missing from the answer.
  • Not collecting like terms, or leaving the answer out of descending order.
  • Dropping a leading coefficient (e.g. losing the \(2\) in \((2x-1)\)).
⇩ Practice Questions ⇩

Confident expanding triple brackets? The natural next step is the reverse skill: factorising quadratic expressions.

Next: Factorising Quadratics

Expanding Triple Brackets: Practice Room

Practise expanding triple brackets with auto-marked questions across four rooms, from single and double brackets up to full triple and quartic expansions. Difficulty rises through the rooms, and every answer is tracked through your global streak system.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Type your expanded expression, for example x^2+5x+6 (use ^ for powers). Term order and spacing do not matter; like terms are collected automatically when your answer is marked. Expand fully and simplify before checking.