Removing brackets 2

Introduction

In this leaflet we show the correct procedure for writing expressions of the form ( a + b )( c + d ) in an alternative form without brackets.

Expressions of the form ( a + b )( c + d ) In the expression ( a + b )( c + d ) it is intended that each term in the first bracket multiplies each\par term in the second.

( a + b )( c + d ) = ac + bc + ad + bd

Example

Removing the brackets from (5 + a )(2 + b ) gives 5 × 2 + a × 2 + 5 × b + a × b which simplifies to 10 + 2 a + 5 b + ab

Example

Removing the brackets from ( x + 6)( x + 2) gives x × x + 6 × x + x × 2 + 6 × 2 which equals x + 6 x + 2 x + 12 which simplifies to x + 8 x + 12

Example

Removing the brackets from ( x + 7)( x - 3) gives x × x + 7 × x + x ×-3 + 7 × 3 which equals x + 7 x - 3 x - 21 which simplifies to

x + 4 x - 21

Example

Removing the brackets from (2 x + 3)( x + 4) gives 2 x × x + 3 × x + 2 x × 4 + 3 × 4 which equals 2 x + 3 x + 8 x + 12 which simplifies to 2 x + 11 x + 12. Occasionally you will need to square a bracketed expression. This can lead to errors. Study the following example.

Example

Remove the brackets from ( x + 1).

Solution

You need to be clear that when a quantity is squared it is multiplied by itself. So ( x + 1) means ( x + 1)( x + 1). Then removing the brackets gives x × x + 1 × x + x × 1 + 1 × 1 which equals x + x + x + 1 which simplifies to x + 2 x + 1 Note that ( x + 1) 2 is not equal to x + 1, and more generally ( x + y ) is not equal to x + y .

Exercises

Remove the brackets from each of the following expressions simplifying your answers where appropriate.

1. a) ( x + 2)( x + 3), b) ( x - 4)( x + 1), c) ( x - 1), d) (3 x + 1)(2 x - 4).

2. a) (2 x -7)( x - 1), b) ( x + 5)(3 x - 1), c) (2 x + 1), d)( x - 3).

Answers

1. a) x + 5x + 6, b) x - 3 x - 4, c) x - 2 x + 1, d) 6x-10 x - 4.\par 2. a) 2 x - 9 x + 7, b) 3 x + 14 x -5, c) 4 x + 4 x + 1, d) x - 6 x + 9.