【典型例题 1】约分：
(1)$\dfrac{-15x^{2}y}{10xy^{2}}$；(2)$\dfrac{a^{2}+2a+1}{a^{2}-1}$；
(3)$\dfrac{2n^{2}-m}{2mn-4n^{3}}$；(4)$\dfrac{xy^{2}+2y}{y}$.
【解】(1)$\dfrac{-15x^{2}y}{10xy^{2}}= -\dfrac{5xy\cdot3x}{5xy\cdot2y}= -\dfrac{3x}{2y}$.
(2)$\dfrac{a^{2}+2a+1}{a^{2}-1}= \dfrac{(a+1)^{2}}{(a+1)(a-1)}= \dfrac{a+1}{a-1}$.
(3)$\dfrac{2n^{2}-m}{2mn-4n^{3}}= \dfrac{-(m-2n^{2})}{2n(m-2n^{2})}= -\dfrac{1}{2n}$.
(4)$\dfrac{xy^{2}+2y}{y}= \dfrac{y(xy+2)}{y}= xy+2$.
规律方法 1. 当分式的分子或分母的系数是负数时,可以运用分式的基本性质把负号约去或把负号提到分式的前面.
2. 注意发现分式的分子和分母的一些隐含的公因式(如互为相反数的式子).
3. 约分一定要彻底,要约到分子、分母没有公因式为止.

(1)
$\begin{aligned}\dfrac{-15x^{2}y}{10xy^{2}}&= -\dfrac{15x^{2}y÷5xy}{10xy^{2}÷5xy}\\&= -\dfrac{3x}{2y}\end{aligned}$
(2)
$\begin{aligned}\dfrac{a^{2}+2a + 1}{a^{2}-1}&=\dfrac{(a + 1)^{2}}{(a + 1)(a-1)}\\&=\dfrac{a + 1}{a-1}\end{aligned}$
(3)
$\begin{aligned}\dfrac{2n^{2}-m}{2mn-4n^{3}}&=\dfrac{-(m - 2n^{2})}{2n(m - 2n^{2})}\\&=-\dfrac{1}{2n}\end{aligned}$
(4)
$\begin{aligned}\dfrac{xy^{2}+2y}{y}&=\dfrac{y(xy + 2)}{y}\\&=xy + 2\end{aligned}$

1. 化简下列分式：
(1)$\dfrac{-35a^{4}b}{21a^{2}b^{3}}$；(2)$\dfrac{x^{2}+6x+9}{x^{2}-9}$.

(1)
解：
$\dfrac{-35a^{4}b}{21a^{2}b^{3}} = \dfrac{-35 ÷ 7 \cdot a^{4-2} \cdot b^{1-3}}{21÷7}$
$= -\dfrac{5a^{2}}{3b^{2}}$
(2)
解：
首先对分子分母进行因式分解
$x^{2} + 6x + 9 = (x + 3)^{2}$
$x^{2} - 9 = (x + 3)(x - 3)$
所以，
$\dfrac{x^{2}+6x+9}{x^{2}-9} = \dfrac{(x + 3)^{2}}{(x + 3)(x - 3)} = \dfrac{x + 3}{x - 3}$

【典型例题 2】通分：
(1)$\dfrac{3}{4a^{2}b}与\dfrac{1}{2ac^{2}}$；
(2)$\dfrac{x+2}{2x+2}与\dfrac{3}{8-4x}$.
【解】(1)分母$4a^{2}b$,$2ac^{2}的最简公分母是4a^{2}bc^{2}$,$\dfrac{3}{4a^{2}b}= \dfrac{3× c^{2}}{4a^{2}b\cdot c^{2}}= \dfrac{3c^{2}}{4a^{2}bc^{2}}$；
$\dfrac{1}{2ac^{2}}= \dfrac{1× 2ab}{2ac^{2}\cdot2ab}= \dfrac{2ab}{4a^{2}bc^{2}}$.
(2)$\dfrac{3}{8-4x}= -\dfrac{3}{4x-8}= -\dfrac{3}{4(x-2)}$,$\dfrac{x+2}{2x+2}= \dfrac{x+2}{2(x+1)}$,分母$4(x-2)$,$2(x+1)的最简公分母是4(x+1)(x-2)$,$\dfrac{x+2}{2x+2}= \dfrac{(x+2)\cdot2(x-2)}{2(x+1)\cdot2(x-2)}= \dfrac{2(x+2)(x-2)}{4(x+1)(x-2)}$；
$\dfrac{3}{8-4x}= -\dfrac{3}{4x-8}= -\dfrac{3×(x+1)}{4(x-2)\cdot(x+1)}= -\dfrac{3(x+1)}{4(x+1)(x-2)}$.

【解】(1)
最简公分母为 $4a^{2}bc^{2}$。
$\dfrac{3}{4a^{2}b} = \dfrac{3 × c^{2}}{4a^{2}b \cdot c^{2}} = \dfrac{3c^{2}}{4a^{2}bc^{2}}$；
$\dfrac{1}{2ac^{2}} = \dfrac{1 × 2ab}{2ac^{2} \cdot 2ab} = \dfrac{2ab}{4a^{2}bc^{2}}$。
(2)
首先化简第二个分数的分母：
$\dfrac{3}{8 - 4x} = -\dfrac{3}{4x - 8} = -\dfrac{3}{4(x - 2)}$，
$\dfrac{x + 2}{2x + 2} = \dfrac{x + 2}{2(x + 1)}$，
最简公分母为 $4(x + 1)(x - 2)$。
$\dfrac{x + 2}{2x + 2} = \dfrac{(x + 2) \cdot 2(x - 2)}{2(x + 1) \cdot 2(x - 2)} = \dfrac{2(x + 2)(x - 2)}{4(x + 1)(x - 2)}$；
$\dfrac{3}{8 - 4x} = -\dfrac{3}{4(x - 2)} = -\dfrac{3 × (x + 1)}{4(x - 2) \cdot (x + 1)} = -\dfrac{3(x + 1)}{4(x + 1)(x - 2)}$。

2. 分式$\dfrac{3a}{a^{2}-b^{2}}的分母经过通分后变成2(a-b)^{2}(a+b)$,那么分子应变为()
A.$6a(a-b)^{2}(a+b)$
B.$2(a-b)$
C.$6a(a-b)$
D.$6a(a+b)$

C

原分式为$\dfrac{3a}{a^{2}-b^{2}}$，分母$a^{2}-b^{2}$可因式分解为$(a-b)(a+b)$。
通分后的分母为$2(a-b)^{2}(a+b)$，即原分母$(a-b)(a+b)$乘以$2(a-b)$。
根据分式的基本性质，分子需同步乘以相同因式$2(a-b)$，即：
$3a × 2(a-b) = 6a(a-b)$。