Solve y^5-81y

$${ y }^{ 5 } -81y$$

Factor

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

Solution Steps

Factor out y.

$$y\left(y^{4}-81\right)$$

Consider y^{4}-81. Rewrite y^{4}-81 as \left(y^{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).

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

Consider y^{2}-9. Rewrite y^{2}-9 as y^{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).

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

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

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

Evaluate

$$y\left(y^{4}-81\right)$$