Question

rút gọn phân thức \(\left(1+\frac{x}{x^2+1}\right):\left(\frac{1}{x-1}-\frac{2x}{x^3+x-x^2-1}\right)\)

 

Answer 1

\(\left(1+\frac{x}{x^2+1}\right):\left(\frac{1}{x-1}-\frac{2x}{x^3-x^2+x-1}\right)\)   \(ĐKXĐ:x\ne\pm1\)

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

\(=\frac{x^2+x+1}{x^2+1}:\frac{x^2-2x+1}{\left(x^2+1\right)\left(x-1\right)}\)

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

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

Answer 2

 ĐKXĐ : x khác 1

Phân thức = x^2+1+x/x^2+1  : [1/x-1 - 2x/(x-1).(x^2+1)]

 = x^2+x+1/x^2+1 : [x^2+1-2x/(x-1).(x^2+1)]

 = x^2+x+1/x^2+1  : [(x-1)^2/(x-1).(x^2+1)]

 = x^2+x+1/x^2+1 : x-1/x^2+1

 = x^2+x+1/x^2+1 . x^2+1/x-1 = x^2+x+1/x-1

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