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 9x2 - 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 .
