\[{\text{Разложите многочлен }}p\left( x \right) = {x^8} + {x^4} + 1{\text{ на множители}}{\text{.}}\]
\[\begin{array}{l}
{x^8} + {x^4} + 1 = {x^8} + 2{x^4} + 1 - {x^4} = {\left( {{x^4} + 1} \right)^2} - {\left( {{x^2}} \right)^2} = \\
\left( {{x^4} + 1 - {x^2}} \right)\left( {{x^4} + 1 + {x^2}} \right) = \left( {{x^4} - {x^2} + 1} \right)\left( {{x^4} + {x^2} + 1} \right)
\end{array}\]
\[\begin{array}{l}
{x^4} + {x^2} + 1 = {x^4} + 2{x^2} + 1 - {x^2} = {\left( {{x^2} + 1} \right)^2} - {x^2} = \\
\left( {{x^2} + 1 - x} \right) \cdot \left( {{x^2} + 1 + x} \right) = \left( {{x^2} - x + 1} \right) \cdot \left( {{x^2} + x + 1} \right)
\end{array}\]
\[p\left( x \right) = \left( {{x^2} + x + 1} \right)\left( {{x^2} - x + 1} \right)\left( {{x^4} - {x^2} + 1} \right)\]
\[p\left( x \right) = \left( {{x^2} + x + 1} \right)\left( {{x^2} - x + 1} \right)\left( {{x^4} - {x^2} + 1} \right)\]
