Solve x^4+xy^3-x^3y-y^4

$$x ^ { 4 } + x y ^ { 3 } - x ^ { 3 } y - y ^ { 4 }$$

Factor

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

Solution Steps

Do the grouping x^{4}+xy^{3}-x^{3}y-y^{4}=\left(x^{4}+xy^{3}\right)+\left(-x^{3}y-y^{4}\right), and factor out x in the first and -y in the second group.

$$x\left(x^{3}+y^{3}\right)-y\left(x^{3}+y^{3}\right)$$

Factor out common term x^{3}+y^{3} by using distributive property.

$$\left(x^{3}+y^{3}\right)\left(x-y\right)$$

Consider x^{3}+y^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

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

Rewrite the complete factored expression.

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

Evaluate

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