Solve a^22-a^18+a^14-a^10+{a}^6-{a}^4

$${ a }^{ 22 } - { a }^{ 18 } + { a }^{ 14 } - { a }^{ 10 } + { a }^{ 6 } - { a }^{ 4 }$$

Factor

$$\left(a-1\right)\left(a+1\right)a^{4}\left(a^{16}+a^{14}+a^{8}+a^{6}+1\right)$$

Solution Steps

Factor out a^{4}.

$$a^{4}\left(a^{18}-a^{14}+a^{10}-a^{6}+a^{2}-1\right)$$

Consider a^{18}-a^{14}+a^{10}-a^{6}+a^{2}-1. Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{18} and m divides the constant factor -1. One such factor is a^{2}-1. Factor the polynomial by dividing it by this factor.

$$\left(a^{2}-1\right)\left(a^{16}+a^{14}+a^{8}+a^{6}+1\right)$$

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^{16}+a^{14}+a^{8}+a^{6}+1 is not factored since it does not have any rational roots.

$$a^{4}\left(a-1\right)\left(a+1\right)\left(a^{16}+a^{14}+a^{8}+a^{6}+1\right)$$

Evaluate

$$\left(a^{2}-1\right)a^{4}\left(a^{16}+a^{14}+a^{8}+a^{6}+1\right)$$