Bài tập về biến đổi các biểu thức hữu tỉ

Bài tập về biến đổi các biểu thức hữu tỉ.

1. 

a) $(\frac{2x+1}{2x-1}-\frac{2x-1}{2x+1}):\frac{4x}{10x+5}$

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

 = $\frac{8x}{(2x-1)(2x+1)}.\frac{5(2x+1)}{4x}$

 = $\frac{10}{2x-1}$

b) $(\frac{1}{x^2+x}-\frac{2-x}{x+1}):(\frac{1}{x}+x-2)$

 = $[\frac{1}{x(x+1)}+\frac{x-2}{x+1}]:\frac{1+x^2-2x}{x}$

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

 = $\frac{1}{x+1}$

c) $\frac{1}{x-1}-\frac{x^3-x}{x^2+1}.(\frac{1}{x^2-2x+1}+\frac{1}{1-x^2})$

 = $\frac{1}{x-1}-\frac{x^3-x}{x^2+1}.[\frac{1}{(x-1{)}^2}-\frac{1}{(x-1)(x+1)}]$

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

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

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

 = $\frac{(x-1{)}^2}{(x^2+1)(x-1)}$

 = $\frac{x-1}{x^2+1}$

d) $(\frac{x}{x+1}-\frac{x^3-2x^2}{x^3+1}):\frac{x}{x+1}+\frac{2x+1}{x^3+1}$

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

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

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

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

 = $\frac{x^2+4x+2}{x^3+1}$

2. $(\frac{x-2}{x+2}+\frac{6x-4}{x^2-4}):\frac{x+1}{x-2}$

 = $\frac{(x-2{)}^2+6x-4}{(x+2)(x-2)}.\frac{x-2}{x+1}$

 = $\frac{x^2-4x+4+6x-4}{(x+2)(x-2)}.\frac{x-2}{x+1}$

 = $\frac{x^2+2x}{(x+2)(x+1)}$

 = $\frac{x(x+2)}{(x+2)(x+1)}$

 = $\frac{x}{x+1}$