Solve x^3+y^3+x^2-y^2

$$x ^ { 3 } + y ^ { 3 } + x ^ { 2 } - y ^ { 2 }$$

Factor

$$\left(x+y\right)\left(x^{2}-xy+x+y^{2}-y\right)$$

Solution Steps

Consider x^{3}+y^{3}+x^{2}-y^{2} as a polynomial over variable x.

$$x^{3}+x^{2}+y^{3}-y^{2}$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{3} and m divides the constant factor y^{3}-y^{2}. One such factor is x+y. Factor the polynomial by dividing it by this factor.

$$\left(x+y\right)\left(x^{2}-xy+x+y^{2}-y\right)$$

Evaluate

$$x^{3}+x^{2}+y^{3}-y^{2}$$