Solve u^2v^6-8u^2=

$$u ^ { 2 } v ^ { 6 } - 8 u ^ { 2 } =$$

Factor

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

Solution Steps

Factor out u^{2}.

$$u^{2}\left(v^{6}-8\right)$$

Consider v^{6}-8. Rewrite v^{6}-8 as \left(v^{2}\right)^{3}-2^{3}. The difference of cubes can be factored using the rule: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right).

$$\left(v^{2}-2\right)\left(v^{4}+2v^{2}+4\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: v^{2}-2,v^{4}+2v^{2}+4.

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

Evaluate

$$u^{2}\left(v^{6}-8\right)$$