Factorize each of the following expressions : (i)`a^6-b^6` (ii) `a^6+b^6` (iii) `a^7+a b^6`

Explanation

(i) `a^6−b^6`
`=(a^2)^3−(b^2)^3`
By using this formula:
`(a^3−b^3)=(a−b)(a^2+ab+b^2)`
`=(a^2−b^2)[(a^2)^2+a^2b^2+(b^2)^2]`
`=(a+b)(a−b)[(a^2)^2+2a^2b^2+(b^2)^2−a^2b^2]`
`=(a+b)(a−b)[(a^2+b^2)^2−a^2b^2]`
`=(a+b)(a−b)[(a^2+b^2+ab)(a^2+b^2−ab)]`

(ii) `a^6+b^6`
`=(a^2)^3+(b^2)^3`
By using this formula:
`(a^3+b^3)=(a+b)(a^2-ab+b^2)`
`=(a^2+b^2)[(a^2)^2-a^2b^2+(b^2)^2]`
`=(a^2+b^2)[a^4-a^2b^2+b^4]`

(iii) `a^7−ab^6=a(a^6−b^6)`
`=a((a^2)^3-(b^2)^3)`
`=a(a^2+b^2)[(a^2)^2-a^2b^2+(b^2)^2]`
`=a(a^2+b^2)[a^4-a^2b^2+b^4]`