Đáp án:
\(\begin{array}{l}
1,\\
\left( {{x^2} - x + 2} \right).\left( {{x^2} + x + 2} \right)\\
2,\\
\left( {{x^2} - x + 3} \right).\left( {{x^2} + x + 3} \right)\\
3,\\
\left( {{x^2} - x + 4} \right).\left( {{x^2} + x + 4} \right)\\
5,\\
\left( {{x^4} - {x^2} + 2} \right).\left( {{x^4} + {x^2} + 2} \right)\\
6,\\
\left( {{x^4} - {x^2} + 3} \right).\left( {{x^4} + {x^2} + 3} \right)
\end{array}\)
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
1,\\
{x^4} + 3{x^2} + 4\\
= \left( {{x^4} + 4{x^2} + 4} \right) - {x^2}\\
= \left[ {{{\left( {{x^2}} \right)}^2} + 2.{x^2}.2 + {2^2}} \right] - {x^2}\\
= {\left( {{x^2} + 2} \right)^2} - {x^2}\\
= \left[ {\left( {{x^2} + 2} \right) - x} \right].\left[ {\left( {{x^2} + 2} \right) + x} \right]\\
= \left( {{x^2} - x + 2} \right).\left( {{x^2} + x + 2} \right)\\
2,\\
{x^4} + 5{x^2} + 9\\
= \left( {{x^4} + 6{x^2} + 9} \right) - {x^2}\\
= \left[ {{{\left( {{x^2}} \right)}^2} + 2.{x^2}.3 + {3^2}} \right] - {x^2}\\
= {\left( {{x^2} + 3} \right)^2} - {x^2}\\
= \left[ {\left( {{x^2} + 3} \right) - x} \right].\left[ {\left( {{x^2} + 3} \right) + x} \right]\\
= \left( {{x^2} - x + 3} \right).\left( {{x^2} + x + 3} \right)\\
3,\\
{x^4} + 7{x^2} + 16\\
= \left( {{x^4} + 8{x^2} + 16} \right) - {x^2}\\
= \left[ {{{\left( {{x^2}} \right)}^2} + 2.{x^2}.4 + {4^2}} \right] - {x^2}\\
= {\left( {{x^2} + 4} \right)^2} - {x^2}\\
= \left[ {\left( {{x^2} + 4} \right) - x} \right].\left[ {\left( {{x^2} + 4} \right) + x} \right]\\
= \left( {{x^2} - x + 4} \right).\left( {{x^2} + x + 4} \right)\\
5,\\
{x^8} + 3{x^4} + 4\\
= \left( {{x^8} + 4{x^4} + 4} \right) - {x^4}\\
= \left[ {{{\left( {{x^4}} \right)}^2} + 2.{x^4}.2 + {2^2}} \right] - {x^4}\\
= {\left( {{x^4} + 2} \right)^2} - {\left( {{x^2}} \right)^2}\\
= \left[ {\left( {{x^4} + 2} \right) - {x^2}} \right].\left[ {\left( {{x^4} + 2} \right) + {x^2}} \right]\\
= \left( {{x^4} - {x^2} + 2} \right).\left( {{x^4} + {x^2} + 2} \right)\\
6,\\
{x^8} + 5{x^4} + 9\\
= \left( {{x^8} + 6{x^4} + 9} \right) - {x^4}\\
= \left[ {{{\left( {{x^4}} \right)}^2} + 2.{x^4}.3 + {3^2}} \right] - {x^4}\\
= {\left( {{x^4} + 3} \right)^2} - {\left( {{x^2}} \right)^2}\\
= \left[ {\left( {{x^4} + 3} \right) - {x^2}} \right].\left[ {\left( {{x^4} + 3} \right) + {x^2}} \right]\\
= \left( {{x^4} - {x^2} + 3} \right).\left( {{x^4} + {x^2} + 3} \right)
\end{array}\)
