Question

Answer 1

\(a,\dfrac{5x+1}{x^3-1}-\dfrac{1-2x}{x^2+x+1}-\dfrac{2}{1-x}\)

\(=\dfrac{5x+1}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{\left(1-2x\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{2\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(=\dfrac{5x+1}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{3x-2x^2-1}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{2x^2+2x+2}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(=\dfrac{5x+1-3x+2x^2+1+2x^2+2x+2}{\left(x-1\right)\left(x^2+x+1\right)}\)

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

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

\(=\dfrac{4}{x-1}\)

\(b,\dfrac{3x^2-x+3}{x^3-1}+\dfrac{1-x}{x^2+x+1}+\dfrac{2}{1-x}\)

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

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

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

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

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

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