Solve {l}{x^{2}+9y^{2}+4z^2-6xy+4xz-12yz}{(p+q)^3-3(p+q)^2(p-q)+3(p+q)(p-q)^2-(p-q)}

$$\left. \begin{array} { l } { x ^ { 2 } + 9 y ^ { 2 } + 4 z ^ { 2 } - 6 x y + 4 x z - 12 y z } \\ { ( p + q ) ^ { 3 } - 3 ( p + q ) ^ { 2 } ( p - q ) + 3 ( p + q ) ( p - q ) ^ { 2 } - ( p - q ) } \end{array} \right.$$

Least Common Multiple

$$\left(x-3y+2z\right)^{2}\left(3pq^{2}-3qp^{2}+7q^{3}+p^{3}+q-p\right)$$

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.

$$\left(x-3y+2z\right)^{2}\left(3pq^{2}-3qp^{2}+7q^{3}+p^{3}+q-p\right)$$

Expand the expression.

$$x^{2}p^{3}+3pq^{2}x^{2}-3qp^{2}x^{2}+7x^{2}q^{3}+qx^{2}-px^{2}+18qxyp^{2}-18pxyq^{2}-42xyq^{3}-6xyp^{3}+6pxy-6qxy+12pxzq^{2}-12qxzp^{2}+4xzp^{3}+28xzq^{3}+4qxz-4pxz+27pq^{2}y^{2}-27qp^{2}y^{2}+9y^{2}p^{3}+63y^{2}q^{3}+9qy^{2}-9py^{2}+36qyzp^{2}-36pyzq^{2}-84yzq^{3}-12yzp^{3}+12pyz-12qyz+12pq^{2}z^{2}-12qp^{2}z^{2}+4z^{2}p^{3}+28z^{2}q^{3}+4qz^{2}-4pz^{2}$$

Evaluate

$$\left(x-3y+2z\right)^{2},\ 3pq^{2}-3qp^{2}+7q^{3}+p^{3}+q-p$$