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【例1】约分：
(1)$-\dfrac{32a^2b^2c}{24b^2cd}$；(2)$\dfrac{x^3 - 2x^2y}{x^2y - 2xy^2}$；(3)$\dfrac{a^2 -9b^2}{a^2 -6ab +9b^2}$
(1)$-\dfrac{32a^2b^2c}{24b^2cd}$；(2)$\dfrac{x^3 - 2x^2y}{x^2y - 2xy^2}$；(3)$\dfrac{a^2 -9b^2}{a^2 -6ab +9b^2}$
答案
解:(1)原式$=-\dfrac{4a^2 · 8b^2c}{3d · 8b^2c}=-\dfrac{4a^2}{3d}$.
(2)原式$=\dfrac{x^2(x-2y)}{xy(x-2y)}=\dfrac{x^2}{xy}=\dfrac{x}{y}$.
(3)原式$=\dfrac{(a+3b)(a-3b)}{(a-3b)^2}=\dfrac{a+3b}{a-3b}$.
(2)原式$=\dfrac{x^2(x-2y)}{xy(x-2y)}=\dfrac{x^2}{xy}=\dfrac{x}{y}$.
(3)原式$=\dfrac{(a+3b)(a-3b)}{(a-3b)^2}=\dfrac{a+3b}{a-3b}$.
【变式1】把分式$\frac{b}{ab+3b}$约分得 （
A.$b+3$
B.$a+3$
C.$\frac{1}{b+3}$
D.$\frac{1}{a+3}$
D
）A.$b+3$
B.$a+3$
C.$\frac{1}{b+3}$
D.$\frac{1}{a+3}$
答案
D
【变式2】在分式$\frac{3b}{3+3a}$，$\frac{a^2+b^2}{a^2-b^2}$，$\frac{m^2-n^2}{m+n}$，$\frac{x^2+xy}{2x}$，$\frac{a+b-c}{c-a-b}$中，最简分式有
1
个。答案
1
【例2】通分：(1)$\frac{b}{3a^2c^2},\frac{c}{-2ab}$与$\frac{a}{5cb^3}$；(2)$\frac{b}{a^2-ab}$与$\frac{a-b}{a^2+ab}$；(3)
答案
解:(1)$\because$最简公分母是$30a^2b^3c^2$,
$\therefore \dfrac{b}{3a^2c^2}=\dfrac{b · 10b^3}{3a^2c^2 · 10b^3}=\dfrac{10b^4}{30a^2b^3c^2}$,
$\dfrac{c}{-2ab}=-\dfrac{c · 15ab^2c^2}{2ab · 15ab^2c^2}=-\dfrac{15ab^2c^3}{30a^2b^3c^2}$,
$\dfrac{a}{5cb^3}=\dfrac{a · 6a^2c}{5cb^3 · 6a^2c}=\dfrac{6a^3c}{30a^2b^3c^2}$.
(2)$\because$最简公分母是$a(a+b)(a-b)$,
$\therefore \dfrac{b}{a^2-ab}=\dfrac{b}{a(a-b)}=\dfrac{b(a+b)}{a(a+b)(a-b)}$,
$\dfrac{a-b}{a^2+ab}=\dfrac{a-b}{a(a+b)}=\dfrac{(a-b)^2}{a(a+b)(a-b)}$.
(3)$\because$最简公分母是$2(x+1)^2(x-1)$,
$\therefore \dfrac{1}{2x+2}=\dfrac{1}{2(x+1)}=\dfrac{(x+1)(x-1)}{2(x+1)^2(x-1)}$;
$\dfrac{3}{x^2-1}=\dfrac{3}{(x+1)(x-1)}=\dfrac{6(x+1)}{2(x+1)^2(x-1)}$;
$\dfrac{x}{x^2+2x+1}=\dfrac{x}{(x+1)^2}=\dfrac{2x(x-1)}{2(x+1)^2(x-1)}$.
$\therefore \dfrac{b}{3a^2c^2}=\dfrac{b · 10b^3}{3a^2c^2 · 10b^3}=\dfrac{10b^4}{30a^2b^3c^2}$,
$\dfrac{c}{-2ab}=-\dfrac{c · 15ab^2c^2}{2ab · 15ab^2c^2}=-\dfrac{15ab^2c^3}{30a^2b^3c^2}$,
$\dfrac{a}{5cb^3}=\dfrac{a · 6a^2c}{5cb^3 · 6a^2c}=\dfrac{6a^3c}{30a^2b^3c^2}$.
(2)$\because$最简公分母是$a(a+b)(a-b)$,
$\therefore \dfrac{b}{a^2-ab}=\dfrac{b}{a(a-b)}=\dfrac{b(a+b)}{a(a+b)(a-b)}$,
$\dfrac{a-b}{a^2+ab}=\dfrac{a-b}{a(a+b)}=\dfrac{(a-b)^2}{a(a+b)(a-b)}$.
(3)$\because$最简公分母是$2(x+1)^2(x-1)$,
$\therefore \dfrac{1}{2x+2}=\dfrac{1}{2(x+1)}=\dfrac{(x+1)(x-1)}{2(x+1)^2(x-1)}$;
$\dfrac{3}{x^2-1}=\dfrac{3}{(x+1)(x-1)}=\dfrac{6(x+1)}{2(x+1)^2(x-1)}$;
$\dfrac{x}{x^2+2x+1}=\dfrac{x}{(x+1)^2}=\dfrac{2x(x-1)}{2(x+1)^2(x-1)}$.
【变式】对$\frac{1}{x-2},\frac{1}{(x-2)(x+3)},\frac{2}{(x+3)^2}$通分的过程中，不正确的是（
A.最简公分母是$(x-2)(x+3)^2$
B.$\frac{1}{x-2}=\frac{(x+3)^2}{(x-2)(x+3)^2}$
C.$\frac{1}{(x-2)(x+3)}=\frac{x+3}{(x-2)(x+3)^2}$
D.$\frac{2}{(x+3)^2}=\frac{2x-2}{(x-2)(x+3)^2}$
D
）A.最简公分母是$(x-2)(x+3)^2$
B.$\frac{1}{x-2}=\frac{(x+3)^2}{(x-2)(x+3)^2}$
C.$\frac{1}{(x-2)(x+3)}=\frac{x+3}{(x-2)(x+3)^2}$
D.$\frac{2}{(x+3)^2}=\frac{2x-2}{(x-2)(x+3)^2}$
答案
D
1. 化简$\frac{a^2}{ab}$的结果是 （
A.$\frac{a}{b}$
B.$\frac{a}{ab}$
C.$\frac{a^2}{b}$
D.$\frac{1}{ab}$
A
）A.$\frac{a}{b}$
B.$\frac{a}{ab}$
C.$\frac{a^2}{b}$
D.$\frac{1}{ab}$
答案
A
2. 下列分式中，是最简分式的是（
A.$\frac{x^2}{x}$
B.$\frac{x^2+1}{y}$
C.$\frac{2}{4x}$
D.$\frac{2-x}{x-2}$
B
）A.$\frac{x^2}{x}$
B.$\frac{x^2+1}{y}$
C.$\frac{2}{4x}$
D.$\frac{2-x}{x-2}$
答案
B
3. 化简 $\frac{x^2 -4}{x^2 -4x +4}$ 的结果是 （
A.$\frac{x-2}{x+2}$
B.$\frac{x+2}{x-2}$
C.$\frac{1}{4x}$
D.$\frac{2-x}{x+2}$
B
）A.$\frac{x-2}{x+2}$
B.$\frac{x+2}{x-2}$
C.$\frac{1}{4x}$
D.$\frac{2-x}{x+2}$
答案
B
4. 分式$\frac{2}{2x-4},\frac{3}{2x}$的最简公分母是 （
A.$2x$
B.$2x(2x-4)$
C.$2x-4$
D.$2x(x-2)$
D
）A.$2x$
B.$2x(2x-4)$
C.$2x-4$
D.$2x(x-2)$
答案
D
5. 下列分式中,能约分的是
①②④
（填序号）.答案
①②④
