Solve 6u^{2}v^{3}+3u^3v^4-9u^5v^6

$$6 u ^ { 2 } v ^ { 3 } + 3 u ^ { 3 } v ^ { 4 } - 9 u ^ { 5 } v ^ { 6 }$$

Factor

$$3u^{2}v^{3}\left(uv-1\right)\left(-3u^{2}v^{2}-3uv-2\right)$$

Solution Steps

Factor out 3.

$$3\left(2u^{2}v^{3}+u^{3}v^{4}-3u^{5}v^{6}\right)$$

Consider 2u^{2}v^{3}+u^{3}v^{4}-3u^{5}v^{6}. Factor out u^{2}v^{3}.

$$u^{2}v^{3}\left(2+uv-3u^{3}v^{3}\right)$$

Consider 2+uv-3u^{3}v^{3}. Consider 2+uv-3u^{3}v^{3} as a polynomial over variable u.

$$-3v^{3}u^{3}+vu+2$$

Find one factor of the form kv^{m}u^{n}+p, where kv^{m}u^{n} divides the monomial with the highest power -3v^{3}u^{3} and p divides the constant factor 2. One such factor is uv-1. Factor the polynomial by dividing it by this factor.

$$\left(uv-1\right)\left(-3u^{2}v^{2}-3uv-2\right)$$

Rewrite the complete factored expression.

$$3u^{2}v^{3}\left(uv-1\right)\left(-3u^{2}v^{2}-3uv-2\right)$$

Evaluate

$$3u^{2}v^{3}\left(uv-3\left(uv\right)^{3}+2\right)$$