Solve x^{2}y-y^{2}z+z^2x-x^2z+y^2x+z^2y-2xyz

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

Factor

$$\left(x-z\right)\left(y-z\right)\left(x+y\right)$$

Solution Steps

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

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

Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power \left(y-z\right)x^{2} and n divides the constant factor yz^{2}-zy^{2}. One such factor is x-z. Factor the polynomial by dividing it by this factor.

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

Consider xy-xz+y^{2}-yz. Do the grouping xy-xz+y^{2}-yz=\left(xy-xz\right)+\left(y^{2}-yz\right), and factor out x in the first and y in the second group.

$$x\left(y-z\right)+y\left(y-z\right)$$

Factor out common term y-z by using distributive property.

$$\left(y-z\right)\left(x+y\right)$$

Rewrite the complete factored expression.

$$\left(x+y\right)\left(x-z\right)\left(y-z\right)$$

Evaluate

$$\left(x-z\right)\left(y-z\right)\left(x+y\right)$$