Find one factor of the form $a^{k}+m$, where $a^{k}$ divides the monomial with the highest power $a^{6}$ and $m$ divides the constant factor $-54$. One such factor is $a^{3}-27$. Factor the polynomial by dividing it by this factor.
$$\left(a^{3}-27\right)\left(a^{3}+2\right)$$
Consider $a^{3}-27$. Rewrite $a^{3}-27$ as $a^{3}-3^{3}$. The difference of cubes can be factored using the rule: $p^{3}-q^{3}=\left(p-q\right)\left(p^{2}+pq+q^{2}\right)$.
$$\left(a-3\right)\left(a^{2}+3a+9\right)$$
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $a^{3}+2,a^{2}+3a+9$.
