Maths Chapter 12: Factorisation
Class 08 Maths NCERTLet 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.
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.
When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions. An irreducible factor is a factor which cannot be expressed further as a product of factors.
A systematic way of factorising an expression is the common factor method. It consists of three steps:
- Write each term of the expression as a product of irreducible factors
- Look for and separate the common factors
- Combine the remaining factors in each term in accordance with the distributive law
Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.
In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.