将 $$$x^{6} - 1$$$ 除以 $$$x^{2} + 1$$$

将题目写成特殊格式（缺失项写为零系数）：

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

步骤 1

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

将计算结果写在表格的上部。

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

从得到的结果中减去被除数：$$$\left(x^{6}-1\right) - \left(x^{6}+x^{4}\right) = - x^{4}-1$$$

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

步骤 2

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

将计算结果写在表格的上部。

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

从得到的结果中减去余数：$$$\left(- x^{4}-1\right) - \left(- x^{4}- x^{2}\right) = x^{2}-1$$$

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

步骤 3

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

将计算结果写在表格的上部。

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

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

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

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：

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

因此，$$$\frac{x^{6} - 1}{x^{2} + 1} = \left(x^{4} - x^{2} + 1\right) + \frac{-2}{x^{2} + 1}$$$。