Factorize each of the following expressions:
(i)`a^4b^4-81 c^4`
(ii) `x y^9-y x^9`
(iii)`x^3-x`
(iv) `18 a^2x^2-32`

Explanation

(i)`a^4b^4-81 c^4`
`=(a^2b^2 + 9c^2)(a^2b^2 - 9c^2)`
`=(ab + 3c)(ab - 3c) `
(ii) `x y^9-y x^9`
`=xy (y^8 - x^8)`
`=xy [(y^4)^2 - (x^4)^2]`
`=xy (y^4 - x^4)(y^4 + x^4)`
`=xy ((y^2)^2 - (x^2)^2)(y^4 + x^4)`
`=xy (y^2 - x^2)(y^2 + x^2)(y^4 + x^4)`
`=xy (y - x)(y + x)(y^2 + x^2)(y^4 + x^4)`

(iii)`x^3-x`
`=x(x^2-1)`
`=x(x-1)(x+1)`

(iv) `18 a^2x^2-32`
`= 2 (9a^2x^2 - 16)`
`= 2 {(3ax)^2 - 4^2}`
`= 2 (3ax + 4) (3ax - 4)`