Polynomials

Exercise 2.5 Part 3

Question 3: Factorise the following using appropriate identities:

(i) `9x^2 + 6xy + y^2`

Solution:

Given, `9x^2 + 6xy + y^2`

`= (3x)^2 + 2 × 3x × y + y^2`

Let, a = 3x and b = y

Using identity, `(a + b) ^2 = a^2 + 2ab + b^2`

`= (3x + y)^2`

`= (3x + y)(3x +y)` Answer

This can also be solved as follows:

`9x^2 + 6xy + y^2`

`= 9x^2 + 3xy + 3xy + y^2`

`= 3x(3x + y) + y(3x + y)`

`= (3x + y)(3x + y)`

(ii) `4y^2 - 4y + 1`

Solution:

Given, `4y^2 - 4y + 1`

`= (2y)^2 - 2xx2yxx1 + 1^2`

Let, a = 2y and b = 1

Using identity, `(a – b)^2 = a^2 - 2ab +b^2`

`= (2y – 1) ^2`

`= (2y – 1)(2y – 1)` Answer

(iii) `x^2-(y^2)/(100)`

Solution: Given: `x^2-(y^2)/(100)`

`=x^2-(y/10)^2`

Let, `a = x` and `b = y/10`

(Using identity `(a^2-b^2)=(a+b)(a-b)`)

The polynomial can be written as follows:

`=(x+y/10)(x-y/10)`