Expanding Triple Brackets
Expanding triple brackets means multiplying out three linear factors, such as (x + a)(x + b)(x + c), to produce a single cubic expression in the form x^3 + ax^2 + bx + c. It is done by expanding two of the brackets first to get a quadratic, then multiplying that quadratic by the remaining bracket and collecting like terms. This is a Higher-tier GCSE algebra skill that builds directly on expanding double brackets.
Before you start
Make sure you're comfortable with these topics first:
Method
- Multiply out any two of the three brackets first, using FOIL or a grid, to get a quadratic expression.
- Collect like terms in the quadratic before moving on.
- Multiply every term of the quadratic by every term in the remaining bracket.
- Collect like terms again, working from the highest power (x^3) down to the constant term.
- Check the term count: a fully expanded triple bracket should simplify to 4 terms (x^3, x^2, x and a constant).
- If a question says 'show that', write out every line of working so the given answer is proven, not just stated.
Worked example
Expand and simplify (x + 2)(x - 3)(x + 4).
- Expand the first two brackets: (x + 2)(x - 3) = x^2 - x - 6.
- Multiply this quadratic by the remaining bracket (x + 4): (x^2 - x - 6)(x + 4).
- Multiply out every term: x^3 + 4x^2 - x^2 - 4x - 6x - 24.
- Collect like terms: x^3 + 3x^2 - 10x - 24.
- The fully simplified answer is x^3 + 3x^2 - 10x - 24.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Expand and simplify (x - 5)(x + 5)(x + 2).
The volume of a box in the shape of a cuboid is given by V = (x + 2)(x + 4)(x - 1) cm^3, where x > 1. (a) Show that V = x^3 + 5x^2 + 2x - 8. (b) Use your calculator to work out the volume when x = 4 cm.
Given that (x + 3)(x + a)(x - 2) = x^3 + 3x^2 - 4x - 12 for all values of x, find the value of a.
Free printable worksheet
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