Question

chứng minh rằng:

\(\dfrac{x+2}{x-1}.\left(\dfrac{x^3}{2x+2}+1\right)-\dfrac{8x+7}{2x^2-2}>0\)

Answer 1

\(=\dfrac{x^3\left(x+2\right)}{2\left(x-1\right)\left(x+1\right)}+\dfrac{x+2}{x-1}-\dfrac{8x+7}{2\left(x-1\right)\left(x+1\right)}\)

\(=\dfrac{x^4+2x^3+2\left(x+1\right)\left(x+2\right)-8x-7}{2\left(x-1\right)\left(x+1\right)}\)

\(=\dfrac{x^4+2x^3+2x^2+6x+4-8x-7}{2\left(x-1\right)\left(x+1\right)}\)

\(=\dfrac{x^4+2x^3+2x^2-2x-3}{2\left(x-1\right)\left(x+1\right)}\)

\(=\dfrac{\left(x-1\right)\left(x+1\right)\left(x^2+2x+3\right)}{2\left(x-1\right)\left(x+1\right)}=\dfrac{x^2+2x+3}{2}>0\)