Solve 1-2a^3+a^6

$$1 - 2 a ^ { 3 } + a ^ { 6 }$$

Factor

$$\left(a-1\right)^{2}\left(a^{2}+a+1\right)^{2}$$

Solution Steps

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 1. One such factor is a^{3}-1. Factor the polynomial by dividing it by this factor.

$$\left(a^{3}-1\right)\left(a^{3}-1\right)$$

Consider a^{3}-1. Rewrite a^{3}-1 as a^{3}-1^{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).

$$\left(a-1\right)\left(a^{2}+a+1\right)$$

Rewrite the complete factored expression. Polynomial a^{2}+a+1 is not factored since it does not have any rational roots.

$$\left(a-1\right)^{2}\left(a^{2}+a+1\right)^{2}$$

Evaluate

$$\left(a^{3}-1\right)^{2}$$