Introduction

Before studying this material you must be familiar with the process of removing brackets. This is because factoring can be thought of as reversing the process of removing brackets. When we factorise an expression it is written as a product of two or more terms, and these will normally involve brackets.

1. Products and Factors

To obtain the product of two numbers they are multiplied together. For example the product of 3 and 4 is 3×4 which equals 12. The numbers which are multiplied together are called factors. We say that 3 and 4 are both factors of 12.

Example

The product of x and y is xy . The product of 5x and 3y is 15xy .

Example

x and 5 y are factors of 10xy since when we multiply x by 5 y we obtain 10 xy .

( x + 1) and ( x + 2) are factors of x + 3 x + 2 because when we multiply ( x +1) by ( x + 2) we obtain x + 3 x + 2.

3 and x -5 are factors of 3 x 15 because 3( x - 5) = 3 x - 15.

2. Common Factors

Sometimes, if we study two expressions to find their factors, we might note that some of the factors are the same. These factors are called common factors .

Example

Consider the numbers 18 and 12.

Both 6 and 3 are factors of 18 because 6 × 3 = 18.

Both 6 and 2 are factors of 12 because 6 × 2 = 12.

So, 18 and 12 share a common factor, namely 6.

In fact 18 and 12 share other common factors. Can you find them?

Example

The number 10 and the expression 15 x share a common factor of 5.

Note that 10 = 5 × 2, and 15 x = 5 × 3 x . Hence 5 is a common factor.

Example

3aand 5a share a common factor of a since

3a = 3 a × a and 5 a = 5 × a . Hence a is a common factor.

Example

8xand 12 x share a common factor of 4 x since

8x = 4x × 2x and 12x = 3x × 4x . Hence 4 x is a common factor.

3. Factoring

To factorize an expression containing two or more terms it is necessary to look for factors which are common to the different terms. Once found, these common factors are written outside a bracketed term. It is ALWAYS possible to check your answers when you factorize by simply removing the brackets again, so you shouldn't get them wrong.

Example

Factorize 15 x + 10.

Solution

First we look for any factors which are common to both 15x and 10. The common factor here is 5. So the original expression can be written

15 x + 10 = 5(3x ) + 5(2)

which shows clearly the common factor. This common factor is written outside a bracketed term, the remaining quantities being placed inside the bracket:

15 x + 10 = 5(3 x + 2)

and the expression has been factorized. We say that the factors of 15 x + 10 are 5 and 3 x + 2. Your answer can be checked by showing 5(3 x + 2) = 5(3 x ) + 5(2) = 15 x + 10

Exercises

Factorize each of the following:

1. 10x + 5y ,

2. 21 + 7x ,

3. xy - 8x ,

4 . 4 x -8xy

Answers

1. 5( 2x + y ),

2. 7(3 + x ),

3. x ( y - 8),

4. 4 x (1 - 2y ).