Question

Answer 1

a: \(\dfrac{x+x^3-3x^2-3}{x-3}\)

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

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

b: \(\dfrac{22x^2+5x^3+10-13x}{5x^2-3x+2}\)

\(=\dfrac{5x^3+22x^2-13x+10}{5x^2-3x+2}\)

\(=\dfrac{5x^3-3x^2+2x+25x^2-15x+10}{5x^2-3x+2}\)

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

c: \(\dfrac{2x^2+x^3+1}{x^2+1}\)

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

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

d: \(\dfrac{-x^3+3x+x^4+x^2}{x^2-2x+3}\)

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

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

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