Double distributivity
To develop a product of two sums
The principle
When expanding the product of two expressions in brackets, such as (a+b)(c+d), each term in the first set of brackets must be multiplied by each term in the second. This is known as double distributivity.
(a+b)(c+d) = a×c + a×d + b×c + b×d
Step-by-step method
- Multiply the first term of the first set of brackets by each term of the second
- Multiply the second term in the first set of brackets by each term in the second
- Add the four results together
- Simplify the expression if any terms can be grouped together
Complete example
Expand (x+2)(x+3):
- x×x = x²
- x×3 = 3x
- 2×x = 2x
- 2×3 = 6
- Sum: x² + 3x + 2x + 6
- Simplified: x² + 5x + 6 (since 3x + 2x = 5x)
Another example with a minus sign
Expand (x-1)(x+4):
- x×x = x²
- x×4 = 4x
- -1×x = -x
- -1×4 = -4
- Sum: x² + 4x - x - 4
- Simplified form: x² + 3x - 4
Common pitfall
The main pitfall is forgetting one of the four products (often the product ‘crossing’ between the second term of the first set of brackets and the first term of the second). To ensure nothing is missed, you can draw arrows linking each term in the first set of brackets to each term in the second: this guarantees that you have indeed obtained four products before simplifying.