Solve yz^2-y^2z-xz^2+x^2z+x{y}^2-{x}^2y

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

Factor

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

Solution Steps

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

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

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

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

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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