将 $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12$$$ 除以 $$$x^{2} - 4 x - 12$$$

按特殊格式书写题目：

$$$\begin{array}{r|r}\hline\\x^{2}- 4 x-12&2 x^{4}- 3 x^{3}- 15 x^{2}+32 x-12\end{array}$$$

步骤 1

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

将计算结果写在表格的上部。

将其乘以除数：$$$2 x^{2} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}$$$。

从得到的结果中减去被除数：$$$\left(2 x^{4}- 3 x^{3}- 15 x^{2}+32 x-12\right) - \left(2 x^{4}- 8 x^{3}- 24 x^{2}\right) = 5 x^{3}+9 x^{2}+32 x-12$$$

$$\begin{array}{r|rrrrr:c}&{\color{Blue}2 x^{2}}&&&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&{\color{Blue}2 x^{4}}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\frac{{\color{Blue}2 x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Blue}2 x^{2}}\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&{\color{Blue}2 x^{2}} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}\\\hline\\&&5 x^{3}&+9 x^{2}&+32 x&-12&\end{array}$$

步骤 2

将所得余式的首项除以除式的首项: $$$\frac{5 x^{3}}{x^{2}} = 5 x$$$

将计算结果写在表格的上部。

将其乘以除数：$$$5 x \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x$$$。

从得到的结果中减去余数：$$$\left(5 x^{3}+9 x^{2}+32 x-12\right) - \left(5 x^{3}- 20 x^{2}- 60 x\right) = 29 x^{2}+92 x-12$$$

$$\begin{array}{r|rrrrr:c}&2 x^{2}&{\color{Peru}+5 x}&&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&2 x^{4}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&\\\hline\\&&{\color{Peru}5 x^{3}}&+9 x^{2}&+32 x&-12&\frac{{\color{Peru}5 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Peru}5 x}\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&{\color{Peru}5 x} \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x\\\hline\\&&&29 x^{2}&+92 x&-12&\end{array}$$

步骤 3

将所得余式的首项除以除式的首项: $$$\frac{29 x^{2}}{x^{2}} = 29$$$

将计算结果写在表格的上部。

将其乘以除数：$$$29 \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348$$$。

从得到的结果中减去余数：$$$\left(29 x^{2}+92 x-12\right) - \left(29 x^{2}- 116 x-348\right) = 208 x+336$$$

$$\begin{array}{r|rrrrr:c}&2 x^{2}&+5 x&{\color{BlueViolet}+29}&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&2 x^{4}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&\\\hline\\&&5 x^{3}&+9 x^{2}&+32 x&-12&\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&\\\hline\\&&&{\color{BlueViolet}29 x^{2}}&+92 x&-12&\frac{{\color{BlueViolet}29 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{BlueViolet}29}\\&&&-\phantom{29 x^{2}}&&&\\&&&29 x^{2}&- 116 x&-348&{\color{BlueViolet}29} \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348\\\hline\\&&&&208 x&+336&\end{array}$$

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：

$$\begin{array}{r|rrrrr:c}&{\color{Blue}2 x^{2}}&{\color{Peru}+5 x}&{\color{BlueViolet}+29}&&&\text{提示}\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&{\color{Blue}2 x^{4}}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\frac{{\color{Blue}2 x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Blue}2 x^{2}}\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&{\color{Blue}2 x^{2}} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}\\\hline\\&&{\color{Peru}5 x^{3}}&+9 x^{2}&+32 x&-12&\frac{{\color{Peru}5 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Peru}5 x}\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&{\color{Peru}5 x} \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x\\\hline\\&&&{\color{BlueViolet}29 x^{2}}&+92 x&-12&\frac{{\color{BlueViolet}29 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{BlueViolet}29}\\&&&-\phantom{29 x^{2}}&&&\\&&&29 x^{2}&- 116 x&-348&{\color{BlueViolet}29} \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348\\\hline\\&&&&208 x&+336&\end{array}$$

因此，$$$\frac{2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12}{x^{2} - 4 x - 12} = \left(2 x^{2} + 5 x + 29\right) + \frac{208 x + 336}{x^{2} - 4 x - 12}$$$。