Expanding triple brackets GCSE questions ask you to multiply three linear expressions together to produce a cubic — an expression with an x³ term. The reliable method is to expand any two brackets first using FOIL, then multiply that quadratic result by the third bracket, collecting like terms at the end.
What does expanding triple brackets mean?
Multiplying two linear brackets, such as (x + 1)(x + 2), produces a quadratic with a highest power of x². Multiplying three linear brackets together produces a cubic — a highest power of x³ — because each of the three brackets can contribute one x to the highest-degree term.
| Brackets multiplied | Resulting degree | Example |
|---|---|---|
| Two (double brackets) | Quadratic (x²) | (x + 1)(x + 2) = x² + 3x + 2 |
| Three (triple brackets) | Cubic (x³) | (x + 1)(x + 2)(x + 3) = x³ + 6x² + 11x + 6 |
Recognising this pattern before you start warns you what the final answer should look like, so a missing x³ term is an instant sign something has gone wrong.
How do you expand three brackets step by step?
- Expand any two of the three brackets first, using FOIL, to produce a quadratic expression.
- Leave the third bracket unexpanded for now.
- Multiply every term of the quadratic by every term of the remaining bracket.
- Collect like terms, grouping by power of x — x³, x², x¹, then the constant.
- Write the final answer with powers in descending order.
Worked example: expand (x + 1)(x + 2)(x + 3).
- Expand the first two brackets: (x + 1)(x + 2) = x² + 3x + 2.
- Multiply the result by the third bracket: (x² + 3x + 2)(x + 3).
- x² × x = x³; x² × 3 = 3x²; 3x × x = 3x²; 3x × 3 = 9x; 2 × x = 2x; 2 × 3 = 6.
- Collect: x³ + 3x² + 3x² + 9x + 2x + 6 = x³ + 6x² + 11x + 6.
Verification: at x = 1, the original brackets give (2)(3)(4) = 24; the expanded answer gives 1 + 6 + 11 + 6 = 24 ✓.
How do you expand a triple bracket with negative terms?
The method is identical, but each negative sign must travel with its term through every multiplication.
Worked example: expand (x − 2)(x + 1)(x − 4).
- Expand the first two brackets: (x − 2)(x + 1) = x² − x − 2.
- Multiply by the third bracket: (x² − x − 2)(x − 4).
- x² × x = x³; x² × (−4) = −4x²; −x × x = −x²; −x × (−4) = 4x; −2 × x = −2x; −2 × (−4) = 8.
- Collect: x³ − 4x² − x² + 4x − 2x + 8 = x³ − 5x² + 2x + 8.
Verification: at x = 0, the original brackets give (−2)(1)(−4) = 8; the expanded answer gives 0 − 0 + 0 + 8 = 8 ✓.
How do you expand a repeated bracket such as (x + 2)³?
(x + 2)³ means (x + 2)(x + 2)(x + 2) — three identical brackets multiplied together. A common error is to cube each term separately, giving x³ + 8, which is wrong because it ignores every cross-term.
- Expand the first two brackets: (x + 2)(x + 2) = x² + 4x + 4.
- Multiply by the third bracket: (x² + 4x + 4)(x + 2).
- x² × x = x³; x² × 2 = 2x²; 4x × x = 4x²; 4x × 2 = 8x; 4 × x = 4x; 4 × 2 = 8.
- Collect: x³ + 2x² + 4x² + 8x + 4x + 8 = x³ + 6x² + 12x + 8.
Verification: at x = 1, (x + 2)³ = 3³ = 27; the expanded answer gives 1 + 6 + 12 + 8 = 27 ✓.
How can you check your triple-bracket expansion is correct?
Substitute a simple number, such as x = 1 or x = 0, into the original three brackets and into your expanded answer. If both give the same value, your expansion is very likely correct. This check takes seconds and reliably catches a dropped term or a sign error — both common when six or more products must be tracked and collected across a single expansion.
Frequently asked questions
Why does expanding triple brackets give an x³ term?
Each of the three brackets can contribute an x to the highest-degree product, so multiplying x from every bracket gives x × x × x = x³. This is the highest power possible from three linear brackets, which is why a fully expanded triple bracket is described as a cubic expression.
Do you have to expand two brackets first, or can you multiply all three at once?
You could attempt all three brackets in a single step, but that means tracking eight separate products at once, which is easy to get wrong. Expanding two brackets first into a quadratic, then multiplying that quadratic by the third bracket, reduces the working to two manageable stages and is the method examiners expect to see shown.
How do you expand (x + 2)³ correctly?
Treat it as (x + 2)(x + 2)(x + 2), never as cubing each term individually. Expand the first pair of brackets to get x² + 4x + 4, then multiply that quadratic by the remaining (x + 2) bracket and collect like terms, which gives x³ + 6x² + 12x + 8.
What is a common mistake when expanding triple brackets?
The most frequent error is losing a term during the second multiplication stage, since a three-term quadratic multiplied by a two-term bracket produces six separate products that all need collecting. Writing out every product methodically, rather than combining steps mentally, is the most reliable way to avoid dropping one and arriving at the wrong cubic.
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