将 $$$u^{5}$$$ 除以 $$$u^{3} + 1$$$

将题目写成特殊格式（缺失项写为零系数）：

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

步骤 1

将被除式的首项除以除式的首项: $$$\frac{u^{5}}{u^{3}} = u^{2}$$$.

将计算结果写在表格的上部。

将其乘以除数：$$$u^{2} \left(u^{3}+1\right) = u^{5}+u^{2}$$$。

从得到的结果中减去被除数：$$$\left(u^{5}\right) - \left(u^{5}+u^{2}\right) = - u^{2}$$$

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

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：

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

因此，$$$\frac{u^{5}}{u^{3} + 1} = u^{2} + \frac{- u^{2}}{u^{3} + 1}$$$。