Understanding and applying simple distributivity

What is expansion?

Expanding an expression means transforming a product (of multiplied factors) into a sum or a difference (of added or subtracted terms), whilst removing any brackets. It is the inverse operation of factorisation.

The distributive law

For any numbers k, a and b:

k(a+b) = k×a + k×b k(a-b) = k×a - k×b

In other words, we multiply k by each term inside the brackets, then add (or subtract) the results.

Examples

Expression	Expansion	Result
3(x+5)	3×x + 3×5	3x + 15
2(x-4)	2×x - 2×4	2x - 8
5(2x+1)	5×2x + 5×1	10x + 5

Step-by-step method

1. Identify the factor outside the brackets (here, k)
2. Multiply k by the first term inside the brackets
3. Multiply k by the second term inside the brackets, retaining the sign between them
4. Write the resulting sum, without brackets

Detailed example: expand 4(x+3)

• 4×x = 4x
• 4 × 3 = 12
• Result: 4x + 12

Common pitfall

The most common pitfall is forgetting to multiply ALL the terms inside the brackets. In 3(x+5), you must not write 3x+5 (you have forgotten to multiply 3 by 5) but rather 3x+15. Always check that each term has been multiplied by the outer factor.