将 $$$x^{4}$$$ 除以 $$$x - 1$$$

将题目写成特殊格式（缺失项写为零系数）：

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

步骤 1

将被除式的首项除以除式的首项: $$$\frac{x^{4}}{x} = x^{3}$$$.

将计算结果写在表格的上部。

将其乘以除数：$$$x^{3} \left(x-1\right) = x^{4}- x^{3}$$$。

从得到的结果中减去被除数：$$$\left(x^{4}\right) - \left(x^{4}- x^{3}\right) = x^{3}$$$

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

步骤 2

将所得余式的首项除以除式的首项: $$$\frac{x^{3}}{x} = x^{2}$$$

将计算结果写在表格的上部。

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

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

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

步骤 3

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

将计算结果写在表格的上部。

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

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

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

步骤 4

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

将计算结果写在表格的上部。

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

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

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

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：

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

因此，$$$\frac{x^{4}}{x - 1} = \left(x^{3} + x^{2} + x + 1\right) + \frac{1}{x - 1}$$$。