Expanding three brackets
Expanding triple brackets is a key skill in algebra that helps us simplify complex expressions. This technique is often used in areas like physics and engineering to model real-world scenarios, such as calculating the volume of irregular shapes or solving equations involving multiple variables. Jump to the questions
Practise now
Each question can be expanded into the form: ax³ + bx² + cx + d
Find the values of a, b, c and d
Topic guide
What this worksheet practises
This worksheet focuses on expanding three brackets multiplied together, such as (x+1)(x+2)(x+3). This is a higher-level algebraic skill that results in a cubic expression (containing x³). It requires careful organisation and systematic working to avoid losing terms.
Key method
You cannot expand three brackets all at once. You must break the problem down into two manageable stages.
- Ignore the first bracket entirely for a moment.
- Expand the second and third brackets using the standard FOIL method.
- Simplify this result by collecting the like terms (the 'x' terms in the middle). Put large brackets around this new quadratic expression.
- Bring down the first bracket you ignored earlier.
- Now, multiply the two terms in your first bracket by the three terms in your new large bracket. This will create six separate terms.
- Collect all the like terms (x³, x², x, and numbers) to find your final cubic expression.
Worked example
Expand and simplify (x + 2)(x + 3)(x + 4).
Step 1: Expand the last two brackets: (x + 3)(x + 4).
x² + 4x + 3x + 12 = x² + 7x + 12.
Step 2: Bring down the first bracket to multiply against this new expression.
(x + 2)(x² + 7x + 12).
Step 3: Multiply 'x' by everything in the second bracket.
x³ + 7x² + 12x.
Step 4: Multiply '2' by everything in the second bracket.
+ 2x² + 14x + 24.
Step 5: Write it all out and collect like terms.
x³ + (7x² + 2x²) + (12x + 14x) + 24
The final answer is x³ + 9x² + 26x + 24.
Common mistakes to avoid
A disastrous mistake is trying to multiply the 'x's together and the numbers together, arriving at an answer like x³ + 24. This skips all the cross-multiplication. You must methodically multiply every term by every other term across the brackets.
Things to remember
If you see a question written as (x + 5)³, do not panic. This simply means (x + 5)(x + 5)(x + 5). Write it out in full immediately, and then apply the exact same two-stage method described above.