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Encapsulate
math

FActoRISAtIon

The process of finding two or more expressions whose product is the given expression is known as factorisation.
When we factorise an algebraic expression, we write it as a product of factors.

Method of factorisation Division of Algebraic Expressions
common factor Method
Breaking down each term to irreducible factors and Division of a monomial by a monomial
taking out the common factors. For example,
22
For example, 133a b c by 19abc = 133ab c = 7ab
2 2
2x + 4 = (2 × x) + (2 × 2) = 2(x + 2) 19abc

Regrouping Method
Regrouping the terms and looking for common Division of a polynomial by a monomial
factors. For example,
2
(
For example, 14x + 28x + 21x by 7x = 72xx + 4x + 3)
2
3
2xy + 3x + 2y + 3 = 2 × x × y + 3 × x + 2 × y + 3 7x
2
= x(2y + 3) + 1 × (2y + 3) = 2x + 4x + 3
= (2y + 3)(x + 1)
Division of a polynomial by a polynomial
Standard Identities For example,
2
• (a + b) = a + 2ab + b 2 5(x + 2)(x + 2)
2
2
2
2
• (a – b) = a – 2ab + b 2 5x + 20x + 20 by (x + 2) = x + 2
• (a – b ) = (a – b) (a + b) = 5(x + 2)
2
2
Splitting the Middle term
We apply this method when the expressions are of the type
2
2
• x + qx + r = x + (a + b)x + ab = (x + a)(x + b) • px + qx + r = (ax + b)(cx + d), where p ≠ 0
2
For example,
2
x + 5x + 6 = (x + 2)(x + 3)
Here a = 2, b = 3, ab = 6, a + b = 5
2
10x + x – 3 = (5x + 3)(2x – 1)
Here, a = 5, b = 3, c = 2 and d = –1 and p = ac = 10, q = bc + ad = 6 – 5 = 1, r = bd = –3

Mental Maths

Split the middle term to factorise the expressions in the left column and then match with the factor of right column.
2
1. x + x – 72 (a) (x – 3)(2x + 3)
2. 2x – 3x – 9 (b) (x + 4)(3x + 2)
2
2
3. 7x + 35x + 42 (c) (2x + 3y)(7x – 5y)
2
4. 3x + 14x + 8 (d) (x + 9)(x – 8)
2
5. 14x + 11xy – 15y (e) 7(x + 2)(x + 3)

307 Factorisation