Page 88 - Maths Skills - 8

86 Maths

INTRODUCTION
You have already learnt about factors and factorization in the previous classes. Factorization or factoring is the
process by which the factors of a composite number are determined and the number is written as a product of these
factors. Just like composite numbers, algebraic expressions also have factors. In this chapter, we will learn about
factorization of a given algebraic expression.

FACTORS

You know that a composite number is a number that can be written as a product of two positive integers other
than 1 and the number itself. For example, 10 is a composite number because it can be written as 5 times 2
(i.e., 10 = 5 × 2).
A composite algebraic expression is similar as it can be written as a product of two or more algebraic expressions.
For example, x + 7x + 12 is a composite algebraic expression because it can be written as (x + 4) (x + 3),
2
i.e., x + 7x + 12 = (x + 4)(x + 3).
2
Recall, the identity, (x + a)(x + b) = x + (a + b)x + ab.
2
In general, a number is a factor of another number if the first number can divide the second number without any
remainder. Similarly, an algebraic expression is a factor of another algebraic expression if the first expression
can divide the second expression without any remainder.
For example, 2xy = 1 × 2 × x × y = (2x) × y = (2y) × x = 2 × (xy) = 1 × 2xy

So, the possible factors of 2xy are 1, 2, x, y, 2x, 2y, xy and 2xy.

FACTORISATION
The process of finding two or more algebraic expressions whose product is the given expression is called the
process of factorisation. In fact it is the reverse process of multiplication.

Now, we have understood the terms factors and factorisation let us learn the different methods of factorisation.

FACTORISATION BY METHOD OF COMMON FACTORS

This method of factorisation breaks down an algebraic expression by taking out those factors which are common
in each term of the algebraic expression.

For example, let us factorise: a b c + abc – a b c
2 2 2
2 2
Here we have, a b c +abc – a b c = abc(abc + 1 – ab)
2 2
2 2 2
Here, the factor abc is common to all the terms of the given algebraic expression.
Hence, the factors of a b c + abc – a b c are abc and (abc + 1 – ab).
2 2 2
2 2
Let us learn more through examples.

Let’s Attempt

Example 1: Factorise each of the following algebraic expressions:

(i) 3x + 18 (ii) 5n + 16n (iii) 7x y – 14x y (iv) 9abc – 3a b
2
3 3
2
2
Solution: (i) 3x + 18 (ii) 5n + 16n (iii) 7x y – 14x y (iv) 9abc – 3a b
3 3
2
2
2
= 3 (x + 6) = n (5n + 16) = 7x y (1 – 2xy ) = 3ab (3c – a)
2
2