**Unit 5 **

**Factors **

**Section 1 Arithmetical factors **

If one number divides exactly into a second number , the first is a factor
**of the second , and the second is a multiple of the first. **

The fraction

24 8

is normally written as

3 1

. It is normal to express fraction

**In their lowest terms. Here 2, 4 and 8 are all factors of both the **
**numerator and the denominator , but 8 is the highest common factor **
**(H. C. F.) A factor which is also a prime number ( 1,2 3,5,7,11,etc ) is **
**called a prime factor. **

**The smallest number which is exactly divisibie by two or more number **
**is called their lowest common multiple ( L. C. M. ). The L. C. M. of 24 and **
36 is 72 .

*Practice 1 *

Write down the answers tob the following : 1. What are the prime factors of thirty-eight ?

2. What is the highest common factor of eighteen and twenty-six ? 3. What is the lowest common multiple of six and eight ?

4. Express the fraction fourteen over twenty-one in its lowest terms.

**Section 2 Algebraic factors**

**If the expression 3x(3x – 5 ) is expanced , we obtain the result 9x**2 - 15x.
If the expression 9x2** - 15x is factorised , we reverse the process and **
obtain the result 3x (3x – 5 ) .

### EXAMPLE

Factorise the following expression : Ax – ay + bx – by

An algebraic expression that is made up of three terms
e.g. 12x2** + 13x – 4 is called a trinomial . **

A trinomial is the product of two binomials ; e.g. ( a + 5 ) ( a – 2 ) = a2 + 3a – 10

and so the factors of a trinomial can be expressed as two binomials.

*Practice 2 *

Expand the following :

1. Three x minus four all squared .

Factorise the following :

1. x squared plus two xy plus y squared .

2. nine a squared plus eighteen ab plus nine b squared . 3. thirty-six minus sixteen a squared .

### Vocabulary practice

Fill in the blank spaces in the following sentences :

1. Twenty-three has only two ________________ , itself and ____________ , and is therefore a __________________ number .

2. To express a fraction or algebraic expression in its ___________________ ____________ , we must divide its terms by the _____________________ ______________________________ .

3. The factors of a ____________ can be expressed as the _______________ of two binomials .