Factorization, Using Identities - Part 2

Factorization using Identities:- Part II

We have used the standard identities to find the product of expression. Now, we shall learn how to use the identities to factorize the given expression.

We shall be using the following three standard identities to factorize expression:-

a. (a + b)² = a² + 2ab + b²

b. ( a – b)² = a² – 2ab + b²

c. a² – b² = (a + b)(a – b)

While using identities for factorization, we need to compare the given expression with the standard identities. If the right hand side of the given expression is similar to the right hand side of any of the identities, then the expression corresponding to the left hand side of that identity is the required factorization. This method has been illustrated through the following examples.

i. 9x² – 6x + 1

Since, the given expression has three terms, it is not comparable with identity ©. Comparing the expression with identities (a) and (b), we can see that the two terms of the expression are perfect squares and its middle term is negative.

Therefore, it is in the form of the identity (b) with a = 3x (since square root of 9x²  = 3x) and b =1,

It means,

a² – 2ab + b² = (3x)² – 2(3x)(1) + (1)²

Hence, the expression can be written in the form (a – b)² as

9x2 – 6x + 1 = (3x – 1)²

                      = (3x – 1)(3x – 1)

Which is required factorization.   

b. 25x² + 30x + 9

Sol: The given expression has three terms. Let us compare it with the identities, we can see that it is in the form of identities (i), how,

Here, a = 5x (√25x² = 5x ) and b = 9 (√9 = 3)

Therefore, a² +2ab + b² = (5x)² + 2.(5x)(3)²

Since, a² + 2ab + b² = (a + b)²

We have,

25x² + 30x + 9 = (5x + 3)²

                          = (5x + 3)(5x + 3)

That is required factorization.

c. 81 a² – 4b²

Let us compare the expression with identities,

we get that it is in the form of

 a² – b²= a² – 2ab +b²

a² – b² = (9a)² – (2b)²

Since,  a² – b² = (a –b)(a + b)

So, (9a)² – (2b)² = (9a +2b)(9a – 2b), Required factorization.

d. (3x + 2y)² – (2 +a – 2b)²

It’s in the form of identities (iii),

Here we have, a = (3x + 2y) and b = (2 + a – 2b)

Since, a2 – b2 = (a + b)(a – b)

So, (3x + 2y)² – (2 +a – 2b)² = {(3x + 2y) + (2 + a – 2b)}{(3x +2y) – (2 + a – 2b)}

= {3x + 2y + 2 + a – 2b}{3x + 2y – 2 – a + 2b}

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