Solve 8/9x^{3}y^{2}-4/3x^{2}y^{3}+8xy^{3}z+2/4xy^{4}+32xy^2z^2-32/3x^2y^2z

$$\frac { 8 } { 9 } x ^ { 3 } y ^ { 2 } - \frac { 4 } { 3 } x ^ { 2 } y ^ { 3 } + 8 x y ^ { 3 } z + \frac { 2 } { 4 } x y ^ { 4 } + 32 x y ^ { 2 } z ^ { 2 } - \frac { 32 } { 3 } x ^ { 2 } y ^ { 2 } z$$

Factor

$$\frac{xy^{2}\left(4x-3y-24z\right)^{2}}{18}$$

Solution Steps

Factor out \frac{1}{18}.

$$\frac{16x^{3}y^{2}-24x^{2}y^{3}+144xy^{3}z+9xy^{4}+576xy^{2}z^{2}-192x^{2}y^{2}z}{18}$$

Consider 16x^{3}y^{2}-24x^{2}y^{3}+144xy^{3}z+9xy^{4}+576xy^{2}z^{2}-192x^{2}y^{2}z. Factor out xy^{2}.

$$xy^{2}\left(16x^{2}-24xy+144yz+9y^{2}+576z^{2}-192xz\right)$$

Consider 16x^{2}-24xy+144yz+9y^{2}+576z^{2}-192xz. Use the perfect square formula, a^{2}+b^{2}+c^{2}+2ab+2ac+2bc=\left(a+b+c\right)^{2}, where a=4x, b=-3y and c=-24z.

$$\left(4x-3y-24z\right)^{2}$$

Rewrite the complete factored expression.

$$\frac{xy^{2}\left(4x-3y-24z\right)^{2}}{18}$$

Evaluate

$$\frac{x\left(y\left(4x-3y-24z\right)\right)^{2}}{18}$$