5.9. a) Factorize $x^{8}+x^{4}+1$ into two factors.

b) Factorize $x^{8}+x^{4}+1$ into four factors, allowing square roots of natural numbers as coefficients.

## 5.2. Proof of identities
5.9. a) $\left(x^{4}+x^{2}+1\right)\left(x^{4}-x^{2}+1\right)$.

b) Each of the resulting fourth-degree polynomials can be factored using the identity

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

by setting $a=1$ and $a=\sqrt{3}$.