Question

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

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

3. \(\dfrac{x^3+4x^2+x-6}{x^3+2x^2-x-2}\)

4. \(\dfrac{ab+cx+ax+bx}{ay+2cx+2ax+cy}\)

Answer 1

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

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

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

\(=\dfrac{x^2-2x+4-x^2-4x-4}{x+2}=\dfrac{-6x}{x+2}\)

3: \(=\dfrac{x^3+3x^2+x^2+3x-2x-6}{x^3+2x^2-x-2}\)

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

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