1. Chứng minh các cặp phân thức sau bằng nhau:
a) $\frac{3}{2x-3}$ và $\frac{3x+6}{2x^{2}+x-6}$
b) $\frac{2}{x+4}$ và $\frac{2x^{2}+6x}{x^{3}+7x^{2}+12x}$
c) $\frac{x^{2}-5x+4}{x^{3}-1}=\frac{x-4}{x^{2}+x+1}$
d) $\frac{x^{2}-3x+2}{4-x^{2}}=\frac{1-x}{x+2}$
2. Rút gọn các phân thức sau:
a) $\frac{y(2x-x^{2})}{x(2y+y^{2})}$
b) $\frac{xy^{3}-yx^{3}}{x^{2}+xy}$
c) $\frac{(x+a)^{2}-x^{2}}{a^{2}+4x^{2}+4ax}$
d) $\frac{(x+a)^{2}-4x^{2}}{a^{2}+9x^{2}+6ax}$
e) $\frac{y(2x-x^{2})(y+2)}{x(2y+y^{2})(x-2)}$
f) $\frac{(xy^{3}-yx^{3})(x-y)}{(x^{2}+xy)(x+y)}$
3. Rút gọn các phân thức sau:
a) $\frac{x^{3}-5x^{2}+6x}{-4x^{2}+10x-4}$
b) $\frac{x^{2}-3xy+2y^{2}}{x^{3}+2x^{2}y-xy^{2}-2y^{3}}$
Bài Làm:
1.
a) $\frac{3}{2x-3}$ = $\frac{3x+6}{2x^{2}+x-6}$
$\Leftrightarrow 3(2x^{2}+x-6)=(2x-3)(3x+6)$
$\Leftrightarrow 6x^{2}+3x-18=6x^{2}-9x+12x-18$
$\Leftrightarrow 6x^{2}+3x-18=6x^{2}+3x-18$
Vậy $\frac{3}{2x-3}$ = $\frac{3x+6}{2x^{2}+x-6}$
b) $\frac{2}{x+4}$ = $\frac{2x^{2}+6x}{x^{3}+7x^{2}+12x}$
$\Leftrightarrow 2(x^{3}+7x^{2}+12x)=(x+4)(2x^{2}+6x)$
$\Leftrightarrow 2x^{3}+14x^{2}+24x=2x^{3}+6x^{2}+8x^{2}+24x$
$\Leftrightarrow 2x^{3}+14x^{2}+24x=2x^{3}+14x^{2}+24x$
Vậy $\frac{2}{x+4}$ = $\frac{2x^{2}+6x}{x^{3}+7x^{2}+12x}$
c) $\frac{x^{2}-5x+4}{x^{3}-1}=\frac{x-4}{x^{2}+x+1}$
$\Leftrightarrow (x^{2}-5x+4)(x^{2}+x+1)=(x^{3}-1)(x-4)$
$\Leftrightarrow (x-4)(x^{3-1})=(x^{3}-1)(x-4)$
Vậy $\frac{x^{2}-5x+4}{x^{3}-1}=\frac{x-4}{x^{2}+x+1}$
d) $\frac{x^{2}-3x+2}{4-x^{2}}=\frac{1-x}{x+2}$
$\Leftrightarrow (x^{2}-3x+2)(x+2)=(1-x)(4-x^{2})$
$\Leftrightarrow x^{3}-3x^{2}+2x+2x^{2}-6x+4=4-x^{2}-4x+x^{3}$
$\Leftrightarrow x^{3}-x^{2}-4x+4=x^{3}-x^{2}-4x+4$
Vậy $\frac{x^{2}-3x+2}{4-x^{2}}=\frac{1-x}{x+2}$
2.
a) $\frac{y(2x-x^{2})}{x(2y+y^{2})}=\frac{yx(2-x)}{xy(2+y)}=\frac{2-x}{2+y}$
b) $\frac{xy^{3}-yx^{3}}{x^{2}+xy}=\frac{xy(y^{2}-x^{2})}{x(x+y)}=\frac{xy(y-x)(y+x)}{x(x+y)}=y(y-x)$
c) $\frac{(x+a)^{2}-x^{2}}{a^{2}+4x^{2}+4ax}=\frac{(x+a-x)(x+a+x)}{(a+2x)^{2}}=\frac{a(a+2x)}{(a+2x)^{2}}=\frac{a}{a+2x}$
d) $\frac{(x+a)^{2}-4x^{2}}{a^{2}+9x^{2}+6ax}=\frac{(x+a)^{2}-(2x)^{2}}{(a+3x)^{2}}=\frac{(x+a-2x)(x+a+2x)}{(a+3x)^{2}}=\frac{(a+3x)(a-x)}{(a+3x)^{2}}=\frac{a-x}{a+3x}$
e) $\frac{y(2x-x^{2})(y+2)}{x(2y+y^{2})(x-2)}=\frac{yx(2-x)(y+2)}{xy(2+y)(x-2)}=1$
f) $\frac{(xy^{3}-yx^{3})(x-y)}{(x^{2}+xy)(x+y)}=\frac{xy(y^{2}-x^{2})(x-y)}{x(x+y)(x+y)}=-\frac{y(x-y)^{2}}{x+y}$
3.
a) $\frac{x^{3}-5x^{2}+6x}{-4x^{2}+10x-4}=\frac{x(x^{2}-5x+6)}{-2(2x^{2}-5x+2)}=\frac{x(x-2)(x-3)}{-2(2x-1)(x-2)}=\frac{x(x-3)}{2(2x-1)}$
b) $\frac{x^{2}-3xy+2y^{2}}{x^{3}+2x^{2}y-xy^{2}-2y^{3}}$
= $\frac{(x-y)(x-2y)}{x^{2}(x+2y)-y^{2}(x+2y)}$
= $\frac{(x-y)(x-2y)}{(x+2y)(x^{2}-y^{2})}$
= $\frac{(x-y)(x-2y)}{(x+2y)(x-y)(x+y)}$
= $\frac{x-2y}{(x+2y)(x+y)}$