📝 题目
例 1 试求 $\frac{1}{{\left( x - 1\right) }^{2}\left( {x - 2}\right) }$ 的部分分式分解.
💡 答案与解析
解 设
$$ \frac{1}{{\left( x - 1\right) }^{2}\left( {x - 2}\right) } = \frac{A}{x - 1} + \frac{{A}^{\prime }}{{\left( x - 1\right) }^{2}} + \frac{B}{x - 2}. $$
消去分母得
$$ 1 = A\left( {{x}^{2} - {3x} + 2}\right) + {A}^{\prime }\left( {x - 2}\right) + B\left( {{x}^{2} - {2x} + 1}\right) . $$
比较上式两边同次项系数得
$$ \begin{array}{ll} {x}^{2} - A & + B = 0, \\ x - {3A} + {A}^{\prime } - {2B} & = 0, \\ {2A} - 2{A}^{\prime } + B & = 1. \end{array} $$
解这方程组得到: $A = - 1,{A}^{\prime } = - 1,B = 1$ . 于是,我们得到
$$ \frac{1}{{\left( x - 1\right) }^{2}\left( {x - 2}\right) } = \frac{-1}{x - 1} + \frac{-1}{{\left( x - 1\right) }^{2}} + \frac{1}{x - 2}. $$