Factorize : (i)`x^3+3x^2+3x-7`
(ii)`x^3-3x^2+3x+7`
(iii) `x^6-7x^3-8`

Explanation

(i) `x^3+3x^2+3x-7`
`=(x+1)^3-2^3`
`a^3−b^3=(a−b)(a^2+ab+b^2)`
`=(x+1-2){(x+1)^2+(x+1)2+2^2}`
`=(x-1){(x^2+1^2+2x+2x+2)+4}`
`=(x-1)(x^2+4x+7)`
`therefore x^3+3x^2+3x-7=(x-1)(x^2+4x+7)`

(ii) `x^3-3x^2+3x+7`
`=x^3−4x^2+x^2+7x−4x+7`
`=(x^3−4x^2+7x)+1(x^2−4x+7)`
`=x(x^2−4x+7)+1(x^2−4x+7)`
`=(x+1)(x^2−4x+7)`
`therefore x^3-3x^2+3x+7=(x+1)(x^2−4x+7)`

(iii) `x^6-7x^3-8`
`=(x^3)2−7(x^3)−8`
`=(x^3−8)(x^3+1)`
`=(x^3−2^3)(x^3+1^3)`
We know that:
`a^3+b^3=(a+b)(a^2−ab+b^2)`
and `a^3−b^3=(a−b)(a^2+ab+b^2)`
`=(x^3−2^3)(x^3+1^3)`
`=(x−2)(x^2+2x+4)(x+1)(x^2−x+1)`
`therefore x^6-7x^3-8=(x−2)(x^2+2x+4)(x+1)(x^2−x+1)`