Solve 81v^2-u^4v^2

$$81 v ^ { 2 } - u ^ { 4 } v ^ { 2 }$$

Factor

$$\left(u-3\right)\left(u+3\right)v^{2}\left(-u^{2}-9\right)$$

Solution Steps

Factor out v^{2}.

$$v^{2}\left(81-u^{4}\right)$$

Consider 81-u^{4}. Rewrite 81-u^{4} as 9^{2}-\left(-u^{2}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(9+u^{2}\right)\left(9-u^{2}\right)$$

Reorder the terms.

$$\left(u^{2}+9\right)\left(-u^{2}+9\right)$$

Consider -u^{2}+9. Rewrite -u^{2}+9 as 3^{2}-u^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(3-u\right)\left(3+u\right)$$

Reorder the terms.

$$\left(-u+3\right)\left(u+3\right)$$

Rewrite the complete factored expression. Polynomial u^{2}+9 is not factored since it does not have any rational roots.

$$v^{2}\left(u^{2}+9\right)\left(-u+3\right)\left(u+3\right)$$

Evaluate

$$v^{2}\left(81-u^{4}\right)$$