The property
The distributive property says:
a(b + c) = ab + ac
The number outside the brackets is "distributed" to every term inside.
Example: 3(4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27. Check: 3 × 9 = 27 ✓.
Picture it: the area model
A rectangle 3 wide and (4 + 5) long can be split into two smaller rectangles:
| × | 4 | 5 |
|---|---|---|
| 3 | 12 | 15 |
Total area = 12 + 15 = 27. The area model shows why distributing works.
Mental maths
- 7 × 102 = 7 × 100 + 7 × 2 = 700 + 14 = 714
- 6 × 49 = 6 × 50 − 6 × 1 = 300 − 6 = 294
- 4 × 3.25 = 4 × 3 + 4 × 0.25 = 12 + 1 = 13
Expanding in algebra
To expand, multiply every term inside by the term outside.
| Expression | Expanded |
|---|---|
| 4(x + 3) | 4x + 12 |
| 5(2y − 1) | 10y − 5 |
| −2(a + 7) | −2a − 14 |
| −3(m − 4) | −3m + 12 |
| x(x + 6) | x² + 6x |
| 2a(3a − b) | 6a² − 2ab |
Distributing a negative
The sign goes along. −(x − 5) = −x + 5. A minus in front of brackets changes every sign inside.
Working backwards: factoring
To factor means to reverse the distributive property. Find the greatest common factor of the terms and pull it outside.
- 6x + 9 = 3(2x + 3)
- 10a − 15b = 5(2a − 3b)
- x² + 4x = x(x + 4)
Check by expanding.
Worked examples
Example 1: Expand 6(n − 2).
6n − 12.
Example 2: Expand −4(3x + 1).
−12x − 4.
Example 3: Factor 8y + 20.
GCF 4 → 4(2y + 5).
Common mistakes
- Multiplying only the first term: 4(x + 3) ≠ 4x + 3.
- Sign errors with negatives outside the brackets.
- Factoring out a number that isn’t the greatest common factor: 12x + 18 = 2(6x + 9) is correct but not fully factored; 6(2x + 3) is.
Try it yourself
- Expand 7(a + 4).
- Expand −5(2x − 3).
- Expand y(y − 9).
- Factor 15m − 10.
- Use the distributive property to find 8 × 97 mentally.
Answers: 1) 7a + 28 2) −10x + 15 3) y² − 9y 4) 5(3m − 2) 5) 800 − 24 = 776
Quick recap
- Distributive property: a(b + c) = ab + ac.
- Expand by multiplying every term inside.
- Factor by taking out the greatest common factor.