Rewrite $x^{6}-64$ as $\left(x^{3}\right)^{2}-8^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x^{3}-8\right)\left(x^{3}+8\right)$$
Consider $x^{3}-8$. Rewrite $x^{3}-8$ as $x^{3}-2^{3}$. The difference of cubes can be factored using the rule: $a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right)$.
$$\left(x-2\right)\left(x^{2}+2x+4\right)$$
Consider $x^{3}+8$. Rewrite $x^{3}+8$ as $x^{3}+2^{3}$. The sum of cubes can be factored using the rule: $a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right)$.
$$\left(x+2\right)\left(x^{2}-2x+4\right)$$
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $x^{2}-2x+4,x^{2}+2x+4$.
