Solve {l}{(x+y+z)^{3}-(y+z-x)^{3}-(z+x-y)^{3}}{7y+3y^{2}z+3z^2x+3yz^2+3yz^2+3zx^2}

$$\left. \begin{array} { l } { ( x + y + z ) ^ { 3 } - ( y + z - x ) ^ { 3 } - ( z + x - y ) ^ { 3 } } \\ { 7 y + 3 y ^ { 2 } z + 3 z ^ { 2 } x + 3 y z ^ { 2 } + 3 y z ^ { 2 } + 3 z x ^ { 2 } } \end{array} \right.$$

Least Common Multiple

$$lcm(2x\left(3\left(y+z\right)^{2}+x^{2}\right)-\left(x-y+z\right)^{3},3xz^{2}+6yz^{2}+7y+3zx^{2}+3zy^{2})$$

Solution Steps

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

$$21zy^{3}+21yzx^{2}+7yz^{3}-7yx^{3}-21y^{2}z^{2}-7y^{4}-126xzy^{2}-21xyz^{2}-21xy^{3}-21x^{2}y^{2}-12zy^{2}x^{3}-3zy^{5}-12zx^{2}y^{3}-3zx^{5}+3z^{2}y^{4}+6z^{2}x^{4}-9yzx^{4}-69yz^{2}x^{3}+6yz^{5}-45yx^{2}z^{3}-15y^{2}z^{4}+9y^{3}z^{3}-9xzy^{4}-75xz^{2}y^{3}+3xz^{5}-27xyz^{4}-108xy^{2}z^{3}-6x^{2}z^{4}-9x^{2}y^{2}z^{2}$$

Expand the expression.

$$-9x^{2}y^{2}z^{2}-21x^{2}y^{2}-6x^{2}z^{4}-21xy^{3}-108xy^{2}z^{3}-27xyz^{4}-21xyz^{2}+3xz^{5}-75xz^{2}y^{3}-9xzy^{4}-126xzy^{2}-7y^{4}+9y^{3}z^{3}-15y^{2}z^{4}-21y^{2}z^{2}-7yx^{3}-45yx^{2}z^{3}+6yz^{5}+7yz^{3}-69yz^{2}x^{3}-9yzx^{4}+21yzx^{2}+6z^{2}x^{4}+3z^{2}y^{4}-3zx^{5}-12zx^{2}y^{3}-3zy^{5}+21zy^{3}-12zy^{2}x^{3}$$

Evaluate

$$2x\left(3\left(y+z\right)^{2}+x^{2}\right)-\left(x-y+z\right)^{3},\ 3xz^{2}+6yz^{2}+7y+3zx^{2}+3zy^{2}$$