Factorization is the process of expressing an algebraic expression as a product of its factors.

For example:
( 6x + 9 = 3(2x + 3) )

Here, we have taken out the common factor 3.

1. Factorisation by Common Factors

In this method, we look for a term that divides all the parts of the expression.

Example:
Factorise ( 8x + 12 )

Solution:
The common factor of 8 and 12 is 4.
So,
( 8x + 12 = 4(2x + 3) )

2. Factorization by Grouping

This is used when there are four terms that can be grouped in pairs.

Example:
Factorize ( ax + ay + bx + by )

Solution:
Group them as ( (ax + ay) + (bx + by) )
Take out common factors from each group:
( a(x + y) + b(x + y) = (a + b)(x + y) )

3. Factorisation using the Difference of Two Squares

An expression of the form ( a^2 - b^2 ) can be written as ( (a + b)(a - b) ).

Example:
Factorize ( x^2 - 9 )

Solution:
( x^2 - 9 = (x + 3)(x - 3) )

4. Factorisation of Quadratic Trinomials

Quadratic trinomials are expressions of the form ( ax^2 + bx + c ).
To factorize, find two numbers that multiply to give ( a \times c ) and add to give ( b ).

Example:
Factorize ( x^2 + 5x + 6 )

Solution:
Numbers that multiply to 6 and add to 5 are 2 and 3.
So,
( x^2 + 5x + 6 = (x + 2)(x + 3) )

5. Solving Equations by Factorization

We can solve equations like ( x^2 + 5x + 6 = 0 ) by factorization.

Solution:
( x^2 + 5x + 6 = (x + 2)(x + 3) = 0 )

Hence, ( x + 2 = 0 ) or ( x + 3 = 0 )

∴( x = -2 ) or ( x = -3 )