Đáp án:

\(\begin{array}{l}
e,\\
{x^{10}} + {x^5} + 1 = \left( {{x^2} + x + 1} \right).\left( {{x^8} - {x^7} + {x^5} - {x^4} + {x^3} - x + 1} \right)\\
f,\\
{x^8} + x + 1 = \left( {{x^2} + x + 1} \right).\left( {{x^6} - {x^5} + {x^3} - {x^2} + 1} \right)\\
g,\\
{x^8} + {x^4} + 1 = \left( {{x^2} + x + 1} \right).\left( {{x^6} - {x^5} + {x^3} - x + 1} \right)
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
e,\\
{x^{10}} + {x^5} + 1\\
 = \left( {{x^{10}} - x} \right) + \left( {{x^5} - {x^2}} \right) + \left( {{x^2} + x + 1} \right)\\
 = x.\left( {{x^9} - 1} \right) + {x^2}.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = x.\left[ {{{\left( {{x^3}} \right)}^3} - {1^3}} \right] + {x^2}.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = x.\left( {{x^3} - 1} \right).\left[ {{{\left( {{x^3}} \right)}^2} + {x^3}.1 + {1^2}} \right] + {x^2}.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = x\left( {{x^3} - 1} \right).\left( {{x^6} + {x^3} + 1} \right) + {x^2}.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^3} - 1} \right).\left( {{x^7} + {x^4} + x} \right) + {x^2}.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^3} - 1} \right).\left( {{x^7} + {x^4} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {x - 1} \right).\left( {{x^2} + x + 1} \right).\left( {{x^7} + {x^4} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^2} + x + 1} \right).\left[ {\left( {x - 1} \right).\left( {{x^7} + {x^4} + {x^2} + x} \right) + 1} \right]\\
 = \left( {{x^2} + x + 1} \right).\left( {{x^8} + {x^5} + {x^3} + {x^2} - {x^7} - {x^4} - {x^2} - x + 1} \right)\\
 = \left( {{x^2} + x + 1} \right).\left( {{x^8} - {x^7} + {x^5} - {x^4} + {x^3} - x + 1} \right)\\
f,\\
{x^8} + x + 1\\
 = \left( {{x^8} - {x^2}} \right) + {x^2} + x + 1\\
 = {x^2}.\left( {{x^6} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = {x^2}.\left[ {{{\left( {{x^3}} \right)}^2} - 1} \right] + \left( {{x^2} + x + 1} \right)\\
 = {x^2}.\left( {{x^3} - 1} \right).\left( {{x^3} + 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^3} - 1} \right).\left( {{x^5} + {x^2}} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {x - 1} \right).\left( {{x^2} + x + 1} \right).\left( {{x^5} + {x^2}} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^2} + x + 1} \right).\left[ {\left( {x - 1} \right).\left( {{x^5} + {x^2}} \right) + 1} \right]\\
 = \left( {{x^2} + x + 1} \right).\left( {{x^6} + {x^3} - {x^5} - {x^2} + 1} \right)\\
 = \left( {{x^2} + x + 1} \right).\left( {{x^6} - {x^5} + {x^3} - {x^2} + 1} \right)\\
g,\\
{x^8} + {x^4} + 1\\
 = \left( {{x^8} - {x^2}} \right) + \left( {{x^4} - x} \right) + \left( {{x^2} + x + 1} \right)\\
 = {x^2}.\left( {{x^6} - 1} \right) + x\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = {x^2}.\left[ {{{\left( {{x^3}} \right)}^2} - {1^2}} \right] + x.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = {x^2}.\left( {{x^3} - 1} \right)\left( {{x^3} + 1} \right) + x.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^3} - 1} \right).\left( {{x^5} + {x^2}} \right) + x.\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^3} - 1} \right).\left( {{x^5} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {x - 1} \right).\left( {{x^2} + x + 1} \right)\left( {{x^5} + {x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
 = \left( {{x^2} + x + 1} \right).\left[ {\left( {x - 1} \right)\left( {{x^5} + {x^2} + x} \right) + 1} \right]\\
 = \left( {{x^2} + x + 1} \right).\left( {{x^6} + {x^3} + {x^2} - {x^5} - {x^2} - x + 1} \right)\\
 = \left( {{x^2} + x + 1} \right).\left( {{x^6} - {x^5} + {x^3} - x + 1} \right)
\end{array}\)