Bài tập nhân và chia các phân thức đại số

1. 

a) $\frac{x-2}{x^{3}+8}.\frac{x^{2}+x-2}{x^{3}-4x^{2}+4x}=\frac{x-2}{(x+2)(x^{2}-2x+4)}.\frac{(x-1)(x+2)}{x(x^{2}-4x+4)}$

 = $\frac{(x-2)(x-1)(x+2)}{(x+2)(x^{2}-2x+4).x(x-2)^{2}}$

 = $\frac{x-1}{x(x^{2}-2x+4).(x-2)}

b) $\frac{x^{3}-2x^{2}y+xy^{2}}{2x+y}.\frac{2xy+y^{2}}{x^{3}-xy^{2}}$

 = $\frac{x(x-y)^{2}}{2x+y}.\frac{y(2x+y)}{x(x-y)(x+y)}$

 = $\frac{xy(x-y)^{2}(2x+y)}{x(2x+y)(x-y)(x+y)}$

 = $\frac{y(x-y)}{x+y}$

c) $\frac{6x-2}{x+5}:(3x-1)$

 = $\frac{6x-2}{x+5}:\frac{1}{3x-1}$

 = $\frac{2(3x-1)}{(x+5)(3x-1)}$

 = $\frac{2}{x+5}$

d) $(x^{2}+4x+4):\frac{2x+4}{5x+3}$

 = $(x+2)^{2}.\frac{5x+3}{2(x+2)}$

 = $\frac{(x+2)^{2}(5x+3)}{2(x+2)}$

 = $\frac{(x+2)(5x+3)}{2}$

2. 

a) $\frac{x^{2}-2xy}{x^{2}y}.A=\frac{x^{2}y-4y^{3}}{3xy^{2}}$

 $\Leftrightarrow A = \frac{x^{2}y-4y^{3}}{3xy^{2}}:\frac{x^{2}-2xy}{x^{2}y}$

 $\Leftrightarrow A = \frac{x^{2}y-4y^{3}}{3xy^{2}}.\frac{x^{2}y}{x^{2}-2xy}$

 $\Leftrightarrow A = \frac{x+2y}{3}$

b) $\frac{x-y}{x^{3}+y^{3}}.A=\frac{x^{2}-2xy+y^{2}}{x^{2}-xy+y^{2}}$

 $\Leftrightarrow A = \frac{x^{2}-2xy+y^{2}}{x^{2}-xy+y^{2}}:\frac{x-y}{x^{3}+y^{3}}$

 $\Leftrightarrow A = \frac{x^{2}-2xy+y^{2}}{x^{2}-xy+y^{2}}.\frac{x^{3}+y^{3}}{x-y}$

 $\Leftrightarrow A = x^{2}-y^{2}$

3. $\frac{x^{5}-1}{x^{2}-4}:\frac{x-1}{x+2}$

Ta có: $x^{5}-1=(x^{5}-x^{4})+(x^{4}-x^{3})+(x^{3}-x^{2})+(x^{2}-x)+(x-1)$

              $=x^{4}(x-1)+x^{3}(x-1)+x^{2}(x-1)+x(x-1)+(x-1)$

              $=(x-1)(x^{4}+x^{3}+x^{2}+x+1)$

Do đó: $\frac{x^{5}-1}{x^{2}-4}:\frac{x-1}{x+2}=\frac{(x-1)(x^{4}+x^{3}+x^{2}+x+1)}{(x-2)(x+2)}.\frac{x+2}{x-1}=\frac{x^{4}+x^{3}+x^{2}+x+1}{x-2}$