Factorize each of the following expressions : `x^(12)-y^(12)` (ii) `x^9-y^9`

Explanation

(i) `x^(12)-y^(12)`
`x^(12)-y^(12)=(x^4)^3-(y^4)^3`
`a^3-b^3=(a-b)(a^2+ab+b^2)`
`(x^4)^3-(y^4)^3=(x^4-y^4)(x^8+x^4y^4+y^8)`
`(x^4)^3-(y^4)^3={(x^2)^2-(y^2)^2}(x^8+x^4y^4+y^8)`
`{(a^2-b^2)=(a+b)(a-b)}`
`(x^4)^3-(y^4)^3=(x^2+y^2)(x^2-y^2)(x^8+x^4y^4+y^8)`
`(x^4)^3-(y^4)^3=(x^2+y^2)(x+y)(x-y)(x^8+x^4y^4+y^8)`
`therefore x^(12)-y^(12)=(x^2+y^2)(x+y)(x-y)(x^8+x^4y^4+y^8)`

(ii) `x^9-y^9`
`x^9-y^9=(x^3)^3-(y^3)^3`
`a^3-b^3=(a-b)(a^2+ab+b^2)`
`(x^3)^3-(y^3)^3=(x-y)(x^2+xy+y^2)(x^6+x^3y^3+y^6)`
`therefore x^9-y^9=(x-y)(x^2+xy+y^2)(x^6+x^3y^3+y^6)`