Khi thực hiện các phép tính \(\dfrac{5x^2-4x+7}{x^3-1}+\dfrac{x+1}{x^2+x+1}+\dfrac{4}{1-x}\), ta có kết quả là
\(\dfrac{2\left(x-1\right)}{x^2+x+1}\).\(\dfrac{2\left(x+1\right)}{x^2+x+1}\).\(\dfrac{7x^2+6x+2}{\left(x-1\right)\left(x^2+x+1\right)}\).\(\dfrac{x^2+6x+2}{\left(x-1\right)\left(x^2+x+1\right)}\).Hướng dẫn giải:\(\dfrac{5x^2-4x+7}{x^3-1}+\dfrac{x+1}{x^2+x+1}+\dfrac{4}{1-x}\)
\(=\dfrac{5x^2-4x+7}{x^3-1}+\dfrac{x+1}{x^2+x+1}+\dfrac{-4}{x-1}\)
\(=\dfrac{5x^2-4x+7+\left(x-1\right)\left(x+1\right)-4\left(x^2+x+1\right)}{\left(x-1\right)\cdot\left(x^2+x+1\right)}\)
\(=\dfrac{5x^2-4x+7+x^2-1-4x^2-4x-4}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{2x^2-4x+2}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{2\left(x^2-2x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{2\left(x-1\right)^2}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{2\left(x-1\right)}{x^2+x+1}\).