factor x^3-(x+y)^3
in general
a^3-b^3 = (a-b)(a^2 + a b + b^2)
so
(x-x-y)(x^2+x^2+xy +x^2+2xy+y^2)
-y ( 3x^2+3xy+y^2)
If you multiply out (x+y)^3 and subtract it, you get rid of the x^3 terms, and then factor out y.
x^3 -(x^3 + 3x^2y + 3xy^2 +y^3)
= -y(3x^2 + 3xy + y^2)
The term in parentheses does not factor with integer coefficients.
To factor the expression x^3 - (x+y)^3, we can use the formula for factoring a difference of cubes.
The formula for factoring a difference of cubes is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In this case, a = x and b = x+y. Substituting these values into the formula, we get:
x^3 - (x+y)^3 = (x - (x+y))(x^2 + (x)(x+y) + (x+y)^2)
Simplifying further, we have:
x^3 - (x+y)^3 = (x - x - y)(x^2 + (x)(x+y) + (x+y)^2)
= (-y)(x^2 + x(x+y) + (x+y)^2)
Expanding the terms inside the brackets, we get:
x^3 - (x+y)^3 = (-y)(x^2 + x^2 + xy + x^2 + 2xy + y^2)
= (-y)(x^2 + 3xy + y^2)
So, the factored form of x^3 - (x+y)^3 is:
x^3 - (x+y)^3 = (-y)(x^2 + 3xy + y^2)