# Factorisation Class 8 Notes- Chapter 14

## What are Factors?

An expression can be factorised into the product of its factors. These factors can be algebraic expressions, variables and numbers also.

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### Division of a monomial by another monomial

Division of a monomial by another monomial:

i) Division of 9x2 by 3:

9x2 Ã· 3 = 3(3x2) / 3 = 3x2

ii) Division of 6x2y by 2y:

6x2yÃ·2y = (6x2)y / 2y = 2y(3x2) / 2y = 3x2

### Division of a polynomial by a monomial

A polynomial 2x3 + 4x2 + 6xis divided by monomial 2x as shown below:

(2x3+4x2+6x)2x = 2x3 / 2x + 4x22x + 6x / 2x = x2+ 2x + 3

### Division of a polynomial by a polynomial

Long division method is used to divide a polynomial by a polynomial.
Example:Division of 3x2 + 3x – 5 by (x – 1) is shown below:

To know more about Polynomial Division, visit here.

### Factors of natural numbers

Every number can be expressed in the form of product of prime factors. This is called prime factor form.

Example: Prime factor form of 42 is2 x 3 x 7, where 2, 3 and 7 are factors of 42.

### Algebraic expressions

An algebraic expression is defined as the mathematical expression which consists of variables, numbers, and operations. The values of this expression are the not constant.For example: x + 1, p – q, 3x, 2x+3y, 5a/6b etc.

### Factors of algebraic expressions and factorisation

AnÂ irreducibleÂ factor is a factor which cannot be expressed further as a product of factors.
Algebraic expressionsÂ can be expressed in irreducible form.

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### Method of Common Factors

#### Factorisation by common factors

To factorise an algebraic expression, theÂ highest common factorsÂ are determined.

Example: Algebraic expression -2y2 + 8y can be written as 2y(-y+4), where 2y is the highest common factor in the expression.

#### Factorisation by regrouping terms

In some algebraic expressions, it is not possible that every term has a common factor. Therefore, to factorise those algebraic expressions, terms having common factors areÂ groupedÂ together.

Example:

= 12a + n – na – 12

= 12a-12+n-na

= 12(a-1)-n(a-1)

= (12-n)(a-1)

(12-n) and (a-1)are factors of the expression 12a+n-na-12

### Method of Identities

#### Algebraic identities

The algebraic equations which are true for all values of variables in them are called algebraic identities.

Some of the identities are,

(a+b)2 = a2+ 2ab + b2

(a-b)2= a2– 2ab + b2

(a+b)(a-b)= a2 – b2

### Factorisation using algebraic identities

Algebraic identities can be used for factorisation

Example: (i)9x2 + 12xy + 4y2

= ( 3x)2 + 2 x 3x x 2y+(2y)2

= (3x+4y)2

(ii)4a2 – b2 = (2a-b)(2a+b)