Remember that the sum of tow cubes identity is: [tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]
So, to create our expression, containing at least two variables, that can be factored using the sum of two cubes, we just need to replace [tex]a[/tex] and [tex]b[/tex] with tow monomials with a different variable:
[tex]a=x[/tex] and [tex]b=y[/tex]
Lets replace those values in our identity:
[tex]x^3+y^3[/tex]
Now that we have our expression, lets factor it using the sum of two cubes identity:
[tex]x^3+y^3=(x+y)(x^2-xy+y^2)[/tex]
To verify if the factored form of our expression (right hand side) is equivalent to the original form (left hand side), we are going to expand the right hand side:
[tex]x^3+y^3=(x+y)(x^2-xy+y^2)[/tex]
[tex]x^3+y^3=x^3-x^2y+xy^2+x^2y-xy^2+y^3[/tex]
[tex]x^3+y^3=x^3+x^2y-x^2y+xy^2-xy^2+y^3[/tex]
[tex]x^3+y^3=x^3+y^3[/tex]
Since both sides of the equation are equal, we can conduce that the factored form of our expression is equivalent to the original expression.
