Factorize : (i)`(x+1)^3` (ii) `(x+1)^3+(x-1)^3` (iii) `8(x+y)^3-27(x-y)^3`

Explanation

(i)`(x+1)^3`
`(a+b)^3=a^3+3a^2b+3ab^2+b^3`
`(x+1)^3=(x^3+3x^2+3x+1)`

(ii) `(x+1)^3+(x-1)^3`
`(a+b)^3=a^3+3a^2b+3ab^2+b^3`
and `(a-b)^3=a^3-3a^2b+3ab^2-b^3`
`(x+1)^3+(x-1)^3=(x^3+3x^2+3x+1)+(x^3-3x^2+3x-1)`
`(x+1)^3+(x-1)^3=2x^3+6x=2x(x^2+3)`
(iii) `8(x+y)^3−27(x−y)^3`
`=[2(x+y)]^3−[3(x−y)]^3`
`=[2(x+y)−3(x−y)][(2(x+y))^2+2(x+y)×3(x−y)(3(x−y))^2]`
`=(2x+2y−3x+3y)[4(x^2+2xy+y^2)+6(x^2−y^2)+9(x^2−2xy+y^2)]`
`=(−x+5y)(4x^2+8xy+4y^2+6x^2−6y^2+9x^2−18xy+9y^2)`
`=(−x+5y)(19x^2−10xy+7y^2)`