Đáp án:
\(\left( {{x^2} + x + 1} \right).\left[ {\left( {x - 1} \right)\left( {{x^3} + 1} \right)\left( {{x^2} + x} \right) + 1} \right]\)
Giải thích các bước giải:
\(\begin{array}{l}
{x^8} + {x^7} + 1\\
= \left( {{x^8} - {x^2}} \right) + \left( {{x^7} - x} \right) + \left( {{x^2} + x + 1} \right)\\
= {x^2}.\left( {{x^6} - 1} \right) + x\left( {{x^6} - 1} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^6} - 1} \right).\left( {{x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left[ {{{\left( {{x^3}} \right)}^2} - {1^2}} \right].\left( {{x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^3} - 1} \right)\left( {{x^3} + 1} \right)\left( {{x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {x - 1} \right)\left( {{x^2} + x + 1} \right).\left( {{x^3} + 1} \right)\left( {{x^2} + x} \right) + \left( {{x^2} + x + 1} \right)\\
= \left( {{x^2} + x + 1} \right).\left[ {\left( {x - 1} \right)\left( {{x^3} + 1} \right)\left( {{x^2} + x} \right) + 1} \right]
\end{array}\)
