Rewrite $x^{6}-729$ as $\left(x^{3}\right)^{2}-27^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x^{3}-27\right)\left(x^{3}+27\right)$$
Consider $x^{3}-27$. Rewrite $x^{3}-27$ as $x^{3}-3^{3}$. The difference of cubes can be factored using the rule: $a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right)$.
$$\left(x-3\right)\left(x^{2}+3x+9\right)$$
Consider $x^{3}+27$. Rewrite $x^{3}+27$ as $x^{3}+3^{3}$. The sum of cubes can be factored using the rule: $a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right)$.
$$\left(x+3\right)\left(x^{2}-3x+9\right)$$
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $x^{2}-3x+9,x^{2}+3x+9$.
