$\left[\begin{array}{l}\text { Identity transformations }] \\ {[\text { Factorization }}\end{array}\right]$

Factorize the polynomial $x^{8}+x^{4}+1$ into four factors.
Represent this polynomial as a difference of squares.

## Solution

$$
\begin{aligned}
& x^{8}+x^{4}+1=\left(x^{8}+2 x^{4}+1\right)-x^{4}=\left(x^{4}+1\right)^{2}-\left(x^{2}\right)^{2}=\left(x^{4}+1-x^{2}\right)\left(x^{4}+1+x^{2}\right)= \\
& \left.=\left(\left(x^{2}+1\right)^{2}-3 x^{2}\right)\right)\left(\left(x^{2}+1\right)^{2}-x^{2}\right)=\left(x^{2}-\sqrt{3} x+1\right)\left(x^{2}+\sqrt{3} x+1\right)\left(x^{2}-x+1\right)\left(x^{2}+x+1\right) .
\end{aligned}
$$