ANSWER
[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} + xy + {y}^{2}][/tex]
EXPLANATION
We want to factor:
[tex] {x}^{3} - {y}^{3} [/tex]
completely.
Recall from binomial theorem that:
[tex]( {x - y)}^{3} = {x}^{3} - 3 {x}^{2} y + 3x {y}^{2} - {y}^{3} [/tex]
We make x³-y³ the subject to get:
[tex] {x}^{3} - {y}^{3} = ( {x - y)}^{3} + 3 {x}^{2} y - \:3x {y}^{2}[/tex]
We now factor the right hand side to get;
[tex]{x}^{3} - {y}^{3} = ( {x - y)}^{3} + 3 {x} y(x - y)[/tex]
We factor further to get,
[tex]{x}^{3} - {y}^{3} = (x - y)[( {x - y)}^{2} + 3 {x} y][/tex]
[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} - 2xy + {y}^{2} + 3 {x} y][/tex]
This finally simplifies to:
[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} + xy + {y}^{2}][/tex]
