Solve a^6-2a^4+1=

$$a ^ { 6 } - 2 a ^ { 4 } + 1 =$$

Factor

$$\left(a-1\right)\left(a+1\right)\left(a^{4}-a^{2}-1\right)$$

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^{2}-1. Factor the polynomial by dividing it by this factor.

$$\left(a^{2}-1\right)\left(a^{4}-a^{2}-1\right)$$

Consider a^{2}-1. Rewrite a^{2}-1 as a^{2}-1^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a-1\right)\left(a+1\right)$$

Rewrite the complete factored expression. Polynomial a^{4}-a^{2}-1 is not factored since it does not have any rational roots.

$$\left(a-1\right)\left(a^{4}-a^{2}-1\right)\left(a+1\right)$$

Evaluate

$$a^{6}-2a^{4}+1$$