What is Single Bracket Expansion?
Single bracket expansion is an algebraic process used to remove brackets by multiplying the term outside the bracket by every term inside the bracket.
It is one of the most important foundational topics in algebra because it helps students simplify expressions and solve equations.
The golden rule: Multiply the term outside the bracket by every term inside the bracket. Don't forget the sign.
The distributive law is used during bracket expansion.
The general pattern
For a single bracket expansion:
a(b + c) = ab + ac
a(b − c) = ab − ac
The term a can be a number, a variable, or a combination of both. Let's work through five different types of examples.
Worked examples
Example 1: Number outside the bracket
The simplest case: a plain number multiplies an expression with two terms.
Expand 3(x + 4)
= 3 × x + 3 × 4 (multiply each term)
= 3x + 12 (simplified answer)
Note: The 3 multiplies both the x and the 4 separately.
Example 2: Negative number outside the bracket
Watch how the sign changes when multiplying a negative by a positive and a negative.
Expand −4(2x − 5)
= −4 × 2x + −4 × (−5) (multiply each term)
= −8x + 20 (simplified answer)
Negative × negative = positive, so −4 × −5 = +20.
Example 3: Variable (letter) outside the bracket
The outside term is a variable, so we use index laws when multiplying.
Expand x(x² + 3x − 7)
= x × x² + x × 3x − x × 7 (multiply each term)
= x³ + 3x² − 7x (simplified answer)
x × x² = x³ because you add the indices: 1 + 2 = 3.
Example 4: Coefficient and variable outside the bracket
When both a number and a variable sit outside the bracket, multiply both together with each term inside.
Expand 5x(2x − 3)
= 5x × 2x − 5x × 3 (multiply each term)
= 10x² − 15x (simplified answer)
5 × 2 = 10 for the numbers; x × x = x² for the letters.
Example 5: Three terms inside the bracket
The same rule applies regardless of how many terms are inside — multiply every single one.
Expand 2x(3x² − 4x + 6)
= 2x × 3x² − 2x × 4x + 2x × 6 (multiply each term)
= 6x³ − 8x² + 12x (simplified answer)
Three terms inside means three separate multiplications — none can be skipped.
Common mistakes to avoid
1. Only expanding one term inside (e.g. writing 3(x + 4) = 3x + 4 instead of 3x + 12).
2. Getting the sign wrong when a negative sits outside.
3. Forgetting index laws when multiplying variables (x × x = x², not 2x).
Key Takeaways
- Single bracket expansion uses the distributive law.
- Multiply the term outside the bracket by every term inside.
- Pay close attention to positive and negative signs.
- Simplify your final answer whenever possible.
- Practise regularly to build confidence in algebra.