Solve {l}{(x-y-z)^2}{(x-y+z)(x-y-z)}{(x+y+z)(x-y-z)}{(x+y+z)(x-y+z)}

$$\left. \begin{array} { l } { ( x - y - z ) ^ { 2 } } \\ { ( x - y + z ) ( x - y - z ) } \\ { ( x + y + z ) ( x - y - z ) } \\ { ( x + y + z ) ( x - y + z ) } \end{array} \right.$$

Least Common Multiple

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

Solution Steps

Factor the expressions that are not already factored.

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

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

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

Expand the expression.

$$x^{4}-2x^{2}z^{2}+2xy^{3}+2xyz^{2}+4xzy^{2}-y^{4}-2yx^{3}+2yz^{3}-2yzx^{2}+z^{4}-2zy^{3}$$

Evaluate

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