CBSE Notes Class 8 Maths Chapter 12 Factorisation

1.0Introduction 

The factorisation is a very important mathematical concept and involves writing a number or an algebraic expression in terms of its factors. Factors of any number or expression can divide it without leaving any remainder.

2.0CBSE Class 8 Math Chapter-12 Factorisation - Revision Notes

What is Factorisation?

Factorisation is a process where we can write any algebraic expression as a product of factors. These factors may be numbers, algebraic variables, or algebraic expressions. 

x² + 5 x + 6 = (x + 2)(x + 3)

In the given example, on the left-hand side, we have an unfactorised equation, and on the right-hand side, there are two factors given in the equation in binomial form.

Key Term

• Factor: A number or expression that exactly divides another number or expression.
• Expression: In maths, expressions are numbers, variables, operations, or other components used to describe a particular mathematical relationship. 

3.0Methods of Factorisation 

1. Common Factor Method: In this method, we write each term of a polynomial in its irreducible form and then find the common factor of the polynomial. For example: 

x² + 6x 

x² = x . x 

6x = 6 . x 

The common factor here is x. 

2. Factorising by Regrouping Terms: It is a method in which we group the terms of an expression into pairs, take out common terms from each, and then factor the common binomial expression. For example:

5ab + 10a + 5b + 10 

After regrouping, 

5ab + 5b + 10a + 10 

5b(a+1) + 10(a+1) 

Now factor out (a + 1)

(a + 1)(5b + 10) 

3. Factorisation by Identity 

There are some formulas present in maths that are used to factor out polynomials in the factorisation. There are three identities that will be used in class 8. 

1. (a + b)² 

= (a+b)(a+b)

= a(a+b) + b(a+b)

= a² + ab + ab + b² 

= a² + 2ab + b²

2. (a – b)²

= (a - b)(a - b)

= a(a-b) -b(a-b) 

= a² – ab – ab + b² 

= a² – 2ab + b²

3. (a + b)(a – b)

= a(a-b) +b(a-b) 

= a² – ab + ab – b² 

= a² – b² 

Solved Examples Related to Identities

Example: Expand using identity: x² + 10x + 25

Compare Identity 1 to this question

a² + 2ab + b²  =  x² +  2.(x).5 + 5²

                  = (x + 5)² 

                  = (x + 5)(x + 5)

Example: Expand using identity: x² – 8x + 16

Compare the question with Identity 2 

a² – 2ab + b² = x² – 2.(x).4 + 4² 

                 = (x – 2)² 

                       = (x – 2)(x – 2) 

Example: Expand using identity: x² – 36 

Compare the question with identity 3 

a² – b² = x² – 6²  

         = (x + 6)(x – 6) 

Example: Evaluate using suitable identities.

1. 95²
2. 196² - 144²

Solution:

a) 95² = (100-5)² = (100)²- 21005+(5)2

= 10000 - 1000 + 25 = 925

b) 196² - 144² = (196+144)(196-144)

= (340)(52) = 17680

4. Factors of the form (x + a)(x + b): To solve these kinds of equations, we need to split the middle term of the polynomial in such a way that on doing addition or subtraction, the byproduct is the same as the middle term while doing multiply, the resultant should be equal to the product of the first and third term of the polynomial. 

(x+a)(x+b) 

x(x+b) + a(x+b)

x² + xb + ax +ab 

x² + x(a+b) + ab 

To elaborate more, let’s have a look at the examples below. 

Example: Expand using identity: x² -  4x - 12

= x² - (6x - 2x) - 12 

= x² - 6x + 2x - 12 

= x(x - 6) + 2(x - 6) 

= (x + 2)(x - 6) 

Example: Expand the following equation: (x+5)(x+3)

Solution:

With the help of the above identity, a = 5, and b=3

Put the value of a and b in x² + x(a+b) + ab 

x² + x(5+3) + 5 3

x² + 8x +15

Example: Factories: x²- 4x- 12

= x² - (6x - 2x) - 12 

= x² - 6x + 2x - 12 

= x(x - 6) + 2(x - 6) 

= (x+2)(x - 6) 

Note: Remember to use the symbols carefully, as one mistake can affect the whole answer. 

4.0Division in Algebraic Expressions

In Maths, for division in algebra, we first factorise both numerator and denominator in their irreducible factors, then cancel the common terms from the equation. Different types of division are as follows:

Division of a Monomial by Another Monomial

10x²/5x  = $\frac{10\times x\times x}{5\times x}=2x$

Division of Polynomial by Monomial 

$\frac{2xy+10x}{2x}=\frac{2x(y+5)}{2x}=(y+5)$

Division of Polynomial by Polynomial 

$\frac{30xy+60y}{5y+10}=\frac{30x(y+2)}{5(y+2)}=6x$

5.0Key Features of CBSE Maths Notes Class 8 for Chapter 12

• These notes provide a step-by-step guide to solving the question related to factorisation. 
• The notes consist of essential identities and formulas to solve the questions with ease. 
• Concepts are explained in a manner that encourages analytical and logical thinking. 
• Clear explanations and examples make these notes ideal for you if you are a self-learner.