Question

(x + y)³ − (x − y)³ can be factorized as

Options

• 2y (3x² + y²)

• 2x (3x² + y²)

• 2y (3y² + x²)

• 2x (x²+ 3y²) 

Answer 1

The given expression to be factorized is 

(x + y)³ − (x − y)³ 

Recall the formula for difference of two cubes  `a^3 -b^3 = (a-b)(a^2 +ab +b^2)`

Using the above formula, we have,

`(x+y)^3 - (x-y)^3`

` = {(x+y) - (x-y)} {(x+y)^2 + (x+y).(x-y) + (x-y)^3}`

` = (x+y - x +y)[{(x)^2 + 2x.y + (y)^2} + (x^2 -y^2) + {x^2 -y^2}+{(x)^2 -2x.y +(y)^2}]`

` = 2y(x^2 + 2 xy + y^2 +x^2 - y^2 +x^2 - 2xy + y^2)`

` = 2y (3x^2 + y^2)`

The given expression to be factorized is

 (x + y)³ − (x − y)³ 

Recall the formula for difference of two cubes  `a^3 -b^3 = (a-b)(a^2 +ab +b^2)`

Using the above formula, we have,

 = (x + y)³ − (x − y)³ 

= `{(x+y) - (x-y)}{(x+y)^2 + (x+y).(x-y)+ (x-y)^2}`

`= (x+y - x+y)[{(x)^2 + 2x.y + (y)^2} + (x^2 - y^2)+ {(x)^2 - 2.x.y + (y^2)}]`

`=2y (x^2 + 2xy + y^2 +x^2 - y^2 + x^2 - 2xy +y^2)`

` = 2y (3x^2 + y^2)`