Rewrite $x^{4}-6561$ as $\left(x^{2}\right)^{2}-81^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x^{2}-81\right)\left(x^{2}+81\right)$$
Consider $x^{2}-81$. Rewrite $x^{2}-81$ as $x^{2}-9^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x-9\right)\left(x+9\right)$$
Rewrite the complete factored expression. Polynomial $x^{2}+81$ is not factored since it does not have any rational roots.
