Expanding Three Brackets
Expanding three or more brackets is a key higher-tier algebra skill. It’s used to find the volume of 3D shapes, solve complex equations, and is a fundamental building block for more advanced maths. This guide will show you the clear, step-by-step process to master it.
The Core Method: A Sequential Process
You can’t multiply three brackets all at once. The key is to do it in a sequence: expand the first two brackets, simplify the result, and then multiply that new expression by the third bracket.
Recap: Expanding Two Brackets
The first step always involves expanding two brackets. You can use the FOIL method or the Grid method.
The FOIL Method
For $(x+2)(x+5)$:
- First: $x \times x = x^2$
- Outer: $x \times 5 = 5x$
- Inner: $2 \times x = 2x$
- Last: $2 \times 5 = 10$
Combine: $x^2 + 5x + 2x + 10 = \mathbf{x^2 + 7x + 10}$.
The Grid Method
For $(x+2)(x+5)$:
| × | $x$ | $+2$ |
|---|---|---|
| $x$ | $x^2$ | $2x$ |
| $+5$ | $5x$ | $10$ |
Combine terms: $x^2 + 2x + 5x + 10 = \mathbf{x^2 + 7x + 10}$.
Worked Example
Expand and simplify $(x+1)(x-2)(x+3)$
- Step 1: Expand the first two brackets.
$(x+1)(x-2) = x^2 – 2x + x – 2 = x^2 – x – 2$.Now the problem is: $(x^2 – x – 2)(x+3)$ - Step 2: Multiply the result by the third bracket.
Multiply every term in the first bracket by every term in the second:
$x^2(x+3) = x^3 + 3x^2$
$-x(x+3) = -x^2 – 3x$
$-2(x+3) = -2x – 6$ - Step 3: Combine and simplify all the terms.
$x^3 + 3x^2 – x^2 – 3x – 2x – 6$
$= x^3 + (3x^2 – x^2) + (-3x – 2x) – 6$
Answer: $x^3 + 2x^2 – 5x – 6$
Tutor Insights
🤔 Common Misunderstandings
- Trying to multiply all three brackets at once. It must be done sequentially: expand two, then multiply by the third.
- Not fully understanding “like terms”. You can only combine terms with the exact same variable and power (e.g., $x^2$ and $x^2$, but not $x^2$ and $x$).
📝 Common Exam Mistakes
- Sign errors. This is the most frequent mistake! Forgetting that a negative times a negative is a positive, or dropping a negative sign in your workings.
- Missing terms. Forgetting to multiply every term from the first result by every term in the next bracket.
- Incorrect simplification after expanding.
Practice Questions
- Expand and simplify: $(x+1)(x+2)(x+3)$
- Expand and simplify: $(x-1)(x+2)(x-3)$
- Expand and simplify: $(x+3)^3$
Show Answers
- Working: $(x^2 + 3x + 2)(x+3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6$.
Answer: $x^3 + 6x^2 + 11x + 6$. - Working: $(x^2 + x – 2)(x-3) = x^3 – 3x^2 + x^2 – 3x – 2x + 6$.
Answer: $x^3 – 2x^2 – 5x + 6$. - Working: $(x+3)(x+3)(x+3) = (x^2 + 6x + 9)(x+3) = x^3 + 3x^2 + 6x^2 + 18x + 9x + 27$.
Answer: $x^3 + 9x^2 + 27x + 27$.
