将 $$$u^{4}$$$ 除以 $$$u^{2} + 1$$$

将题目写成特殊格式（缺失项写为零系数）：

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

步骤 1

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

将计算结果写在表格的上部。

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

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

$$\begin{array}{r|rrrrr:c}&{\color{DarkMagenta}u^{2}}&&&&&\\\hline\\{\color{Magenta}u^{2}}+1&{\color{DarkMagenta}u^{4}}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{DarkMagenta}u^{4}}}{{\color{Magenta}u^{2}}} = {\color{DarkMagenta}u^{2}}\\&-\phantom{u^{4}}&&&&&\\&u^{4}&+0 u^{3}&+u^{2}&&&{\color{DarkMagenta}u^{2}} \left(u^{2}+1\right) = u^{4}+u^{2}\\\hline\\&&&- u^{2}&+0 u&+0&\end{array}$$

步骤 2

将所得余式的首项除以除式的首项: $$$\frac{- u^{2}}{u^{2}} = -1$$$

将计算结果写在表格的上部。

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

从得到的结果中减去余数：$$$\left(- u^{2}\right) - \left(- u^{2}-1\right) = 1$$$

$$\begin{array}{r|rrrrr:c}&u^{2}&{\color{BlueViolet}-1}&&&&\\\hline\\{\color{Magenta}u^{2}}+1&u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\\&-\phantom{u^{4}}&&&&&\\&u^{4}&+0 u^{3}&+u^{2}&&&\\\hline\\&&&{\color{BlueViolet}- u^{2}}&+0 u&+0&\frac{{\color{BlueViolet}- u^{2}}}{{\color{Magenta}u^{2}}} = {\color{BlueViolet}-1}\\&&&-\phantom{- u^{2}}&&&\\&&&- u^{2}&+0 u&-1&{\color{BlueViolet}-1} \left(u^{2}+1\right) = - u^{2}-1\\\hline\\&&&&&1&\end{array}$$

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：

$$\begin{array}{r|rrrrr:c}&{\color{DarkMagenta}u^{2}}&{\color{BlueViolet}-1}&&&&\text{提示}\\\hline\\{\color{Magenta}u^{2}}+1&{\color{DarkMagenta}u^{4}}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{DarkMagenta}u^{4}}}{{\color{Magenta}u^{2}}} = {\color{DarkMagenta}u^{2}}\\&-\phantom{u^{4}}&&&&&\\&u^{4}&+0 u^{3}&+u^{2}&&&{\color{DarkMagenta}u^{2}} \left(u^{2}+1\right) = u^{4}+u^{2}\\\hline\\&&&{\color{BlueViolet}- u^{2}}&+0 u&+0&\frac{{\color{BlueViolet}- u^{2}}}{{\color{Magenta}u^{2}}} = {\color{BlueViolet}-1}\\&&&-\phantom{- u^{2}}&&&\\&&&- u^{2}&+0 u&-1&{\color{BlueViolet}-1} \left(u^{2}+1\right) = - u^{2}-1\\\hline\\&&&&&1&\end{array}$$

因此，$$$\frac{u^{4}}{u^{2} + 1} = \left(u^{2} - 1\right) + \frac{1}{u^{2} + 1}$$$。