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【典型例题 1】计算：
(1)$\dfrac{a^{2}+b}{2ab}-\dfrac{a^{2}-b}{2ab}$；
(2)$\dfrac{3a}{b - a}-\dfrac{b}{b - a}-\dfrac{a + b}{b - a}$.
(1)$\dfrac{a^{2}+b}{2ab}-\dfrac{a^{2}-b}{2ab}$；
(2)$\dfrac{3a}{b - a}-\dfrac{b}{b - a}-\dfrac{a + b}{b - a}$.
答案
【解】(1)原式$=\dfrac{a^{2}+b - a^{2}+b}{2ab}= \dfrac{2b}{2ab}= \dfrac{1}{a}$.
(2)原式$=\dfrac{3a - b - a - b}{b - a}= \dfrac{-2(b - a)}{b - a}= -2$.
(2)原式$=\dfrac{3a - b - a - b}{b - a}= \dfrac{-2(b - a)}{b - a}= -2$.
1. 计算：
(1)$\dfrac{3a^{2}-5a}{a^{2}+1}+\dfrac{2a^{2}+4}{a^{2}+1}-\dfrac{2a^{2}-5a + 1}{a^{2}+1}$；
(2)$\dfrac{a - 3b}{a - b}+\dfrac{a + b}{a - b}$.
(1)$\dfrac{3a^{2}-5a}{a^{2}+1}+\dfrac{2a^{2}+4}{a^{2}+1}-\dfrac{2a^{2}-5a + 1}{a^{2}+1}$；
(2)$\dfrac{a - 3b}{a - b}+\dfrac{a + b}{a - b}$.
答案
(1)
$\begin{aligned}&\dfrac{3a^{2}-5a}{a^{2}+1}+\dfrac{2a^{2}+4}{a^{2}+1}-\dfrac{2a^{2}-5a + 1}{a^{2}+1}\\=&\dfrac{3a^{2}-5a + 2a^{2}+4-(2a^{2}-5a + 1)}{a^{2}+1}\\=&\dfrac{3a^{2}-5a + 2a^{2}+4 - 2a^{2}+5a - 1}{a^{2}+1}\\=&\dfrac{(3a^{2}+2a^{2}-2a^{2})+(-5a + 5a)+(4 - 1)}{a^{2}+1}\\=&\dfrac{3a^{2}+3}{a^{2}+1}\\=&\dfrac{3(a^{2}+1)}{a^{2}+1}\\=&3\end{aligned}$
(2)
$\begin{aligned}&\dfrac{a - 3b}{a - b}+\dfrac{a + b}{a - b}\\=&\dfrac{a - 3b+a + b}{a - b}\\=&\dfrac{(a + a)+(-3b + b)}{a - b}\\=&\dfrac{2a-2b}{a - b}\\=&\dfrac{2(a - b)}{a - b}\\=&2\end{aligned}$
$\begin{aligned}&\dfrac{3a^{2}-5a}{a^{2}+1}+\dfrac{2a^{2}+4}{a^{2}+1}-\dfrac{2a^{2}-5a + 1}{a^{2}+1}\\=&\dfrac{3a^{2}-5a + 2a^{2}+4-(2a^{2}-5a + 1)}{a^{2}+1}\\=&\dfrac{3a^{2}-5a + 2a^{2}+4 - 2a^{2}+5a - 1}{a^{2}+1}\\=&\dfrac{(3a^{2}+2a^{2}-2a^{2})+(-5a + 5a)+(4 - 1)}{a^{2}+1}\\=&\dfrac{3a^{2}+3}{a^{2}+1}\\=&\dfrac{3(a^{2}+1)}{a^{2}+1}\\=&3\end{aligned}$
(2)
$\begin{aligned}&\dfrac{a - 3b}{a - b}+\dfrac{a + b}{a - b}\\=&\dfrac{a - 3b+a + b}{a - b}\\=&\dfrac{(a + a)+(-3b + b)}{a - b}\\=&\dfrac{2a-2b}{a - b}\\=&\dfrac{2(a - b)}{a - b}\\=&2\end{aligned}$
【典型例题 2】计算：(1)$\dfrac{7a}{8b^{2}c}-\dfrac{b}{12a^{2}c}$；
(2)$\dfrac{24}{x^{2}-16}+\dfrac{3}{4 - x}$.
(2)$\dfrac{24}{x^{2}-16}+\dfrac{3}{4 - x}$.
答案
思路导引 异分母的分式相加减,先找出最简公分母进行通分,变为同分母的分式相加减,最后结果一定要化成最简分式或整式.
【解】
(1)原式$=\dfrac{21a^{3}}{24a^{2}b^{2}c}-\dfrac{2b^{3}}{24a^{2}b^{2}c}= \dfrac{21a^{3}-2b^{3}}{24a^{2}b^{2}c}$.
(2)原式$=\dfrac{24}{(x + 4)(x - 4)}-\dfrac{3(x + 4)}{(x + 4)(x - 4)}= \dfrac{24 - 3(x + 4)}{(x + 4)(x - 4)}= \dfrac{24 - 3x - 12}{(x + 4)(x - 4)}= \dfrac{-3x + 12}{(x + 4)(x - 4)}= \dfrac{-3(x - 4)}{(x + 4)(x - 4)}= -\dfrac{3}{x + 4}$.
【解】
(1)原式$=\dfrac{21a^{3}}{24a^{2}b^{2}c}-\dfrac{2b^{3}}{24a^{2}b^{2}c}= \dfrac{21a^{3}-2b^{3}}{24a^{2}b^{2}c}$.
(2)原式$=\dfrac{24}{(x + 4)(x - 4)}-\dfrac{3(x + 4)}{(x + 4)(x - 4)}= \dfrac{24 - 3(x + 4)}{(x + 4)(x - 4)}= \dfrac{24 - 3x - 12}{(x + 4)(x - 4)}= \dfrac{-3x + 12}{(x + 4)(x - 4)}= \dfrac{-3(x - 4)}{(x + 4)(x - 4)}= -\dfrac{3}{x + 4}$.
