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Question

How to factorize by using identity

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Solution

Factorization using Identities :There are some identities and using that the factorization is much easier.

A number of expressions to be factorized are of the form or can be put into the form : a
2 + 2ab + b 2 , a 2 – 2ab + b 2 , a 2 – b 2 and x 2 + (a + b) + ab. These expressions can be easily factorized using Identities I, II, III and IV

In this section we will learn Factorization using Identities one by one.


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

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

(3) a2 – b2 = (a + b)(a – b).

4) x2 + (a + b) x + ab = (x + a) (x + b)


Examples:


(i) 4m2 – 12mn + 9n2

Solution:

We can express 4m2 – 12mn + 9n2 as using a2 - 2ab + b2 = (a - b)2

= (2m)2 - 2(2m)(3n) + (3n)2

= (2m – 3n)2

= (2m - 3n)(2m - 3n)

(ii) 16x2 – 36y2

Solution:

We can express 16x2 – 36y2 as using a2 – b2 = (a + b)(a - b).

= (4x)2 - (6y)2

= (4x + 6y)(4x – 6y)


(iii) 1 – 25(2a – 5b)2

Solution:

We can express 1 – 25(2a – 5b)2 as using a2 – b2 = (a + b)(a - b).

= (1)2 - [5(2a – 5b)]2

= [1 + 5(2a – 5b)] [1 - 5(2a – 5b)]

= (1 + 10a – 25b) (1 – 10a + 25b)

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