Sheet № 195 · Higher only · AQA · Edexcel · OCR
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.
§Key definitions
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.
Question:
This is a Higher only topic, but here is a straightforward example. Expand (x + 1)(x + 2)(x + 3).
Answer:
x³ + 6x² + 11x + 6
Q1 (Higher):
Expand and simplify (x + 2)(x + 5)(x + 1).
§Formulas to memorise
(x + a)(x + b)(x + c) = x³ + (a + b + c)x² + (ab + ac + bc)x + abc
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.
Worked example
This is a Higher only topic, but here is a straightforward example. Expand (x + 1)(x + 2)(x + 3).
Working:
⚠ 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.