多项式长除法计算器

按特殊格式书写题目：

$$$\begin{array}{r|r}\hline\\x-7&x^{3}- 12 x^{2}+38 x-17\end{array}$$$

步骤 1

将被除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：$$$x^{2} \left(x-7\right) = x^{3}- 7 x^{2}$$$。

从得到的结果中减去被除数：$$$\left(x^{3}- 12 x^{2}+38 x-17\right) - \left(x^{3}- 7 x^{2}\right) = - 5 x^{2}+38 x-17$$$

$$\begin{array}{r|rrrr:c}&{\color{DeepPink}x^{2}}&&&&\\\hline\\{\color{Magenta}x}-7&{\color{DeepPink}x^{3}}&- 12 x^{2}&+38 x&-17&\frac{{\color{DeepPink}x^{3}}}{{\color{Magenta}x}} = {\color{DeepPink}x^{2}}\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&{\color{DeepPink}x^{2}} \left(x-7\right) = x^{3}- 7 x^{2}\\\hline\\&&- 5 x^{2}&+38 x&-17&\end{array}$$

步骤 2

将所得余式的首项除以除式的首项: $$$\frac{- 5 x^{2}}{x} = - 5 x$$$

将计算结果写在表格的上部。

将其乘以除数：$$$- 5 x \left(x-7\right) = - 5 x^{2}+35 x$$$。

从得到的结果中减去余数：$$$\left(- 5 x^{2}+38 x-17\right) - \left(- 5 x^{2}+35 x\right) = 3 x-17$$$

$$\begin{array}{r|rrrr:c}&x^{2}&{\color{SaddleBrown}- 5 x}&&&\\\hline\\{\color{Magenta}x}-7&x^{3}&- 12 x^{2}&+38 x&-17&\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&\\\hline\\&&{\color{SaddleBrown}- 5 x^{2}}&+38 x&-17&\frac{{\color{SaddleBrown}- 5 x^{2}}}{{\color{Magenta}x}} = {\color{SaddleBrown}- 5 x}\\&&-\phantom{- 5 x^{2}}&&&\\&&- 5 x^{2}&+35 x&&{\color{SaddleBrown}- 5 x} \left(x-7\right) = - 5 x^{2}+35 x\\\hline\\&&&3 x&-17&\end{array}$$

步骤 3

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

将计算结果写在表格的上部。

将其乘以除数：$$$3 \left(x-7\right) = 3 x-21$$$。

从得到的结果中减去余数：$$$\left(3 x-17\right) - \left(3 x-21\right) = 4$$$

$$\begin{array}{r|rrrr:c}&x^{2}&- 5 x&{\color{BlueViolet}+3}&&\\\hline\\{\color{Magenta}x}-7&x^{3}&- 12 x^{2}&+38 x&-17&\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&\\\hline\\&&- 5 x^{2}&+38 x&-17&\\&&-\phantom{- 5 x^{2}}&&&\\&&- 5 x^{2}&+35 x&&\\\hline\\&&&{\color{BlueViolet}3 x}&-17&\frac{{\color{BlueViolet}3 x}}{{\color{Magenta}x}} = {\color{BlueViolet}3}\\&&&-\phantom{3 x}&&\\&&&3 x&-21&{\color{BlueViolet}3} \left(x-7\right) = 3 x-21\\\hline\\&&&&4&\end{array}$$

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：

$$\begin{array}{r|rrrr:c}&{\color{DeepPink}x^{2}}&{\color{SaddleBrown}- 5 x}&{\color{BlueViolet}+3}&&\text{提示}\\\hline\\{\color{Magenta}x}-7&{\color{DeepPink}x^{3}}&- 12 x^{2}&+38 x&-17&\frac{{\color{DeepPink}x^{3}}}{{\color{Magenta}x}} = {\color{DeepPink}x^{2}}\\&-\phantom{x^{3}}&&&&\\&x^{3}&- 7 x^{2}&&&{\color{DeepPink}x^{2}} \left(x-7\right) = x^{3}- 7 x^{2}\\\hline\\&&{\color{SaddleBrown}- 5 x^{2}}&+38 x&-17&\frac{{\color{SaddleBrown}- 5 x^{2}}}{{\color{Magenta}x}} = {\color{SaddleBrown}- 5 x}\\&&-\phantom{- 5 x^{2}}&&&\\&&- 5 x^{2}&+35 x&&{\color{SaddleBrown}- 5 x} \left(x-7\right) = - 5 x^{2}+35 x\\\hline\\&&&{\color{BlueViolet}3 x}&-17&\frac{{\color{BlueViolet}3 x}}{{\color{Magenta}x}} = {\color{BlueViolet}3}\\&&&-\phantom{3 x}&&\\&&&3 x&-21&{\color{BlueViolet}3} \left(x-7\right) = 3 x-21\\\hline\\&&&&4&\end{array}$$

因此，$$$\frac{x^{3} - 12 x^{2} + 38 x - 17}{x - 7} = \left(x^{2} - 5 x + 3\right) + \frac{4}{x - 7}$$$。