将 $$$u^{3}$$$ 除以 $$$u - 1$$$

将题目写成特殊格式（缺失项写为零系数）：

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

步骤 1

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

将计算结果写在表格的上部。

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

从得到的结果中减去被除数：$$$\left(u^{3}\right) - \left(u^{3}- u^{2}\right) = u^{2}$$$

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

步骤 2

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

将计算结果写在表格的上部。

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

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

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

步骤 3

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

将计算结果写在表格的上部。

将其乘以除数：$$$1 \left(u-1\right) = u-1$$$。

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

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

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：

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

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