NCERT Textbook- Factorisation Class 8 Notes | EduRev

Mathematics (Maths) Class 8

Created by: Indu Gupta

Class 8 : NCERT Textbook- Factorisation Class 8 Notes | EduRev

 Page 1


FACTORISATION  217
14.1  Introduction
14.1.1  Factors of natural numbers
Y ou will remember what you learnt about factors in Class VI. Let us take a natural number,
say 30, and write it as a product of other natural numbers, say
 30 = 2 × 15
= 3 × 10 = 5 × 6
Thus, 1, 2, 3, 5, 6, 10, 15 and 30 are the factors of 30.
Of these, 2, 3 and 5 are the prime factors of 30 (Why?)
A number written as a product of prime factors is said to
be in the prime factor form; for example, 30 written as
2 × 3 × 5 is in the prime factor form.
The prime factor form of 70 is 2 × 5 × 7.
The prime factor form of 90 is 2 × 3 × 3 × 5, and so on.
Similarly, we can express algebraic expressions as products of their factors.  This is
what we shall learn to do in this chapter.
14.1.2  Factors of algebraic expressions
W e have seen in Class VII that in algebraic expressions, terms are formed as products of
factors. For example, in the algebraic expression 5xy + 3x the term 5xy has been formed
by the factors 5, x and y, i.e.,
5xy = y x × × 5
Observe that the factors 5, x and y of 5xy cannot further
be expressed as a product of factors. We may say that 5,
x and y are ‘prime’ factors of 5xy.  In algebraic expressions,
we use the word ‘irreducible’ in place of ‘prime’.  W e say that
5 × x × y is the irreducible form of 5xy.  Note 5 × (xy) is not
an irreducible form of  5xy, since the factor xy can be further
expressed as a product of x and y, i.e., xy = x × y.
Factorisation
CHAPTER
14
Note 1 is a factor of 5xy, since
5xy = y x × × × 5 1
In fact, 1 is a factor of every term. As
in the case of natural numbers, unless
it is specially required, we do not show
1 as a separate factor of any term.
We know that 30 can also be written as
30 = 1 × 30
Thus, 1 and 30 are also factors of 30.
You will notice that 1 is a factor of any
number.  For example, 101 = 1 × 101.
However, when we write a number as a
product of factors, we shall not write 1 as
a factor, unless it is specially required.
Page 2


FACTORISATION  217
14.1  Introduction
14.1.1  Factors of natural numbers
Y ou will remember what you learnt about factors in Class VI. Let us take a natural number,
say 30, and write it as a product of other natural numbers, say
 30 = 2 × 15
= 3 × 10 = 5 × 6
Thus, 1, 2, 3, 5, 6, 10, 15 and 30 are the factors of 30.
Of these, 2, 3 and 5 are the prime factors of 30 (Why?)
A number written as a product of prime factors is said to
be in the prime factor form; for example, 30 written as
2 × 3 × 5 is in the prime factor form.
The prime factor form of 70 is 2 × 5 × 7.
The prime factor form of 90 is 2 × 3 × 3 × 5, and so on.
Similarly, we can express algebraic expressions as products of their factors.  This is
what we shall learn to do in this chapter.
14.1.2  Factors of algebraic expressions
W e have seen in Class VII that in algebraic expressions, terms are formed as products of
factors. For example, in the algebraic expression 5xy + 3x the term 5xy has been formed
by the factors 5, x and y, i.e.,
5xy = y x × × 5
Observe that the factors 5, x and y of 5xy cannot further
be expressed as a product of factors. We may say that 5,
x and y are ‘prime’ factors of 5xy.  In algebraic expressions,
we use the word ‘irreducible’ in place of ‘prime’.  W e say that
5 × x × y is the irreducible form of 5xy.  Note 5 × (xy) is not
an irreducible form of  5xy, since the factor xy can be further
expressed as a product of x and y, i.e., xy = x × y.
Factorisation
CHAPTER
14
Note 1 is a factor of 5xy, since
5xy = y x × × × 5 1
In fact, 1 is a factor of every term. As
in the case of natural numbers, unless
it is specially required, we do not show
1 as a separate factor of any term.
We know that 30 can also be written as
30 = 1 × 30
Thus, 1 and 30 are also factors of 30.
You will notice that 1 is a factor of any
number.  For example, 101 = 1 × 101.
However, when we write a number as a
product of factors, we shall not write 1 as
a factor, unless it is specially required.
218  MATHEMATICS
Next consider the expression 3x (x + 2).  It can be written as a product of factors.
3, x and (x + 2)
3x(x + 2) = () 2 3 + × × x x
The factors 3, x and (x +2) are irreducible factors of 3x (x + 2).
Similarly, the expression 10x (x + 2) (y + 3) is expressed in its irreducible factor form
as  10x (x + 2) (y + 3) = ( ) ( ) 25 2 3 xx y × × ×+× + .
14.2  What is Factorisation?
When we factorise an algebraic expression, we write it as a product of factors.  These
factors may be numbers, algebraic variables or algebraic expressions.
Expressions like 3xy, y x
2
5 , 2x (y + 2), 5 (y + 1) (x + 2) are already in factor form.
Their factors can be just read off from them, as we already know .
On the other hand consider expressions like 2x + 4, 3x + 3y, x
2
 + 5x, x
2
 + 5x + 6.
It is not obvious what their factors are. W e need to develop systematic methods to factorise
these expressions, i.e., to find their factors. This is what we shall do now.
14.2.1  Method of common factors
• W e begin with a simple example: Factorise 2x + 4.
W e shall write each term as a product of irreducible factors;
2x =2 × x
4 = 2 × 2
Hence 2x + 4 = (2 × x) + (2 × 2)
Notice that factor 2 is common to both the terms.
Observe, by distributive law
2 × (x + 2) = (2 × x) + (2 × 2)
Therefore, we can write
2x + 4 = 2 × (x + 2) = 2 (x + 2)
Thus, the expression 2x + 4 is the same as 2 (x + 2).  Now we can read off its factors:
they are 2 and (x + 2). These factors are irreducible.
Next, factorise 5xy + 10x.
The irreducible factor forms of 5xy and 10x are respectively,
5xy =5 × x × y
10x =2 × 5 × x
Observe that the two terms have 5 and x as common factors.  Now,
5xy + 10x = (5 × x × y)  + (5 × x × 2)
=(5x × y)  + (5x × 2)
W e combine the two terms using the distributive law ,
(5x× y) + (5x× 2) = 5x × ( y + 2)
Therefore, 5xy + 10x = 5 x (y + 2). (This is the desired factor form.)
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