1. Rút gọn các phân thức sau:
a) ( \(\dfrac{x+1}{x-1}\) - \(\dfrac{x-1}{x+1}\)) : ( \(\dfrac{1}{x+1}\) - \(\dfrac{x}{1-x}\) + \(\dfrac{2}{x^2-1}\))
b) \(\dfrac{2+x}{2-x}\) : \(\dfrac{4x^2}{4-4x+x^2}\).( \(\dfrac{2}{2-x}\) - \(\dfrac{4}{8+x^3}\) . \(\dfrac{4-2x+x^2}{2-x}\))
c) [( \(\dfrac{3}{x-y}\) + \(\dfrac{3x}{x^2-y^2}\)) : \(\dfrac{2x+y}{x^2+2xy+y^2}\)] . \(\dfrac{x-y}{3}\)
a: \(=\dfrac{x^2+2x+1-x^2+2x-1}{\left(x-1\right)\left(x+1\right)}:\left(\dfrac{1}{x+1}+\dfrac{x}{x-1}+\dfrac{2}{\left(x-1\right)\left(x+1\right)}\right)\)
\(=\dfrac{4x}{\left(x-1\right)\left(x+1\right)}:\dfrac{x-1+x^2+x+2}{\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{4x}{\left(x-1\right)\left(x+1\right)}\cdot\dfrac{\left(x-1\right)\left(x+1\right)}{x^2+2x+1}=\dfrac{4x}{x^2+2x+1}\)
b: \(=\dfrac{x+2}{-\left(x-2\right)}\cdot\dfrac{\left(x-2\right)^2}{4x^2}\cdot\left(\dfrac{2}{2-x}-\dfrac{4}{\left(x+2\right)\left(x^2-2x+4\right)}\cdot\dfrac{x^2-2x+4}{2-x}\right)\)
\(=\dfrac{-\left(x+2\right)\left(x-2\right)}{4x^2}\cdot\left(\dfrac{2}{2-x}-\dfrac{4}{\left(x+2\right)\left(2-x\right)}\right)\)
\(=\dfrac{-\left(x+2\right)\left(x-2\right)}{4x^2}\cdot\dfrac{2x+4-4}{\left(2-x\right)\left(x+2\right)}\)
\(=\dfrac{2x}{4x^2}=\dfrac{1}{2x}\)
