将 $$$x^{6}$$$ 除以 $$$\left(x^{2} + 1\right)^{2}$$$

将除数改写为：$$$\left(x^{2} + 1\right)^{2} = x^{4} + 2 x^{2} + 1$$$。

将题目写成特殊格式（缺失项写为零系数）：

$$$\begin{array}{r|r}\hline\\x^{4}+2 x^{2}+1&x^{6}+0 x^{5}+0 x^{4}+0 x^{3}+0 x^{2}+0 x+0\end{array}$$$

步骤 1

将被除式的首项除以除式的首项: $$$\frac{x^{6}}{x^{4}} = x^{2}$$$.

将计算结果写在表格的上部。

将其乘以除数：$$$x^{2} \left(x^{4}+2 x^{2}+1\right) = x^{6}+2 x^{4}+x^{2}$$$。

从得到的结果中减去被除数：$$$\left(x^{6}\right) - \left(x^{6}+2 x^{4}+x^{2}\right) = - 2 x^{4}- x^{2}$$$

$$\begin{array}{r|rrrrrrr:c}&{\color{OrangeRed}x^{2}}&&&&&&&\\\hline\\{\color{Magenta}x^{4}}+2 x^{2}+1&{\color{OrangeRed}x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\frac{{\color{OrangeRed}x^{6}}}{{\color{Magenta}x^{4}}} = {\color{OrangeRed}x^{2}}\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+2 x^{4}&+0 x^{3}&+x^{2}&&&{\color{OrangeRed}x^{2}} \left(x^{4}+2 x^{2}+1\right) = x^{6}+2 x^{4}+x^{2}\\\hline\\&&&- 2 x^{4}&+0 x^{3}&- x^{2}&+0 x&+0&\end{array}$$

步骤 2

将所得余式的首项除以除式的首项: $$$\frac{- 2 x^{4}}{x^{4}} = -2$$$

将计算结果写在表格的上部。

将其乘以除数：$$$- 2 \left(x^{4}+2 x^{2}+1\right) = - 2 x^{4}- 4 x^{2}-2$$$。

从得到的结果中减去余数：$$$\left(- 2 x^{4}- x^{2}\right) - \left(- 2 x^{4}- 4 x^{2}-2\right) = 3 x^{2}+2$$$

$$\begin{array}{r|rrrrrrr:c}&x^{2}&{\color{Blue}-2}&&&&&&\\\hline\\{\color{Magenta}x^{4}}+2 x^{2}+1&x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+2 x^{4}&+0 x^{3}&+x^{2}&&&\\\hline\\&&&{\color{Blue}- 2 x^{4}}&+0 x^{3}&- x^{2}&+0 x&+0&\frac{{\color{Blue}- 2 x^{4}}}{{\color{Magenta}x^{4}}} = {\color{Blue}-2}\\&&&-\phantom{- 2 x^{4}}&&&&&\\&&&- 2 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&-2&{\color{Blue}-2} \left(x^{4}+2 x^{2}+1\right) = - 2 x^{4}- 4 x^{2}-2\\\hline\\&&&&&3 x^{2}&+0 x&+2&\end{array}$$

由于余式的次数小于除式的次数，故除法完成。

所得表格再次显示如下：

$$\begin{array}{r|rrrrrrr:c}&{\color{OrangeRed}x^{2}}&{\color{Blue}-2}&&&&&&\text{提示}\\\hline\\{\color{Magenta}x^{4}}+2 x^{2}+1&{\color{OrangeRed}x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\frac{{\color{OrangeRed}x^{6}}}{{\color{Magenta}x^{4}}} = {\color{OrangeRed}x^{2}}\\&-\phantom{x^{6}}&&&&&&&\\&x^{6}&+0 x^{5}&+2 x^{4}&+0 x^{3}&+x^{2}&&&{\color{OrangeRed}x^{2}} \left(x^{4}+2 x^{2}+1\right) = x^{6}+2 x^{4}+x^{2}\\\hline\\&&&{\color{Blue}- 2 x^{4}}&+0 x^{3}&- x^{2}&+0 x&+0&\frac{{\color{Blue}- 2 x^{4}}}{{\color{Magenta}x^{4}}} = {\color{Blue}-2}\\&&&-\phantom{- 2 x^{4}}&&&&&\\&&&- 2 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&-2&{\color{Blue}-2} \left(x^{4}+2 x^{2}+1\right) = - 2 x^{4}- 4 x^{2}-2\\\hline\\&&&&&3 x^{2}&+0 x&+2&\end{array}$$

因此，$$$\frac{x^{6}}{\left(x^{2} + 1\right)^{2}} = \left(x^{2} - 2\right) + \frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}$$$。