Solve x^4-2x^3y+y^4-2xy^3+2{x}^2{y}^2

$${ x }^{ 4 } -2 { x }^{ 3 } y+ { y }^{ 4 } -2x { y }^{ 3 } +2 { x }^{ 2 } { y }^{ 2 }$$

Factor

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

Solution Steps

Consider x^{4}-2x^{3}y+y^{4}-2xy^{3}+2x^{2}y^{2} as a polynomial over variable x.

$$x^{4}-2yx^{3}+2y^{2}x^{2}-2y^{3}x+y^{4}$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor y^{4}. One such factor is x^{2}+y^{2}. Factor the polynomial by dividing it by this factor.

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

Consider x^{2}-2xy+y^{2}. Use the perfect square formula, a^{2}-2ab+b^{2}=\left(a-b\right)^{2}, where a=x and b=y.

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

Rewrite the complete factored expression.

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

Evaluate

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