Solve v^4w^3-81w^3

$$v ^ { 4 } w ^ { 3 } - 81 w ^ { 3 }$$

Factor

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

Solution Steps

Factor out w^{3}.

$$w^{3}\left(v^{4}-81\right)$$

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

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

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

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

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

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

Evaluate

$$w^{3}\left(v^{4}-81\right)$$