1: \(\dfrac{4x^3-2x^2-3x+1}{x-2}\)
\(=\dfrac{4x^3-8x^2+6x^2-12x+9x-18+19}{x-2}\)
\(=4x^2+6x+9+\dfrac{19}{x-2}\)
2: \(\dfrac{2x^4-x^3-3x^2-2x}{x-2}\)
\(=\dfrac{2x^4-4x^3+5x^3-10x^2+7x^2-14x+12x-24+24}{x-2}\)
\(=2x^3+5x^2+7x+12+\dfrac{24}{x-2}\)
6)\(\dfrac{4x^3-2x^2+1-2x}{2x-1}=\dfrac{\left(2x-1\right)\left(4x^2-2\right)}{2x-1}=4x^2-2\)
5)\(\dfrac{x^3-2x^2-9x+18}{x-3}=\dfrac{\left(x-2\right)\left(x-3\right)\left(x+3\right)}{x-3}=\left(x+3\right)\left(x-2\right)=x^2+x-6\)
3)\(\dfrac{2x^4-x^3-3x^2-2x}{x-2}=\dfrac{\left(x-2\right)\left(2x^3+3x^2+3x+4\right)-8}{x-2}=\)\(\left(2x^3+3x^2+3x+4\right)\)\(-\dfrac{8}{x-2}\)
4)\(\dfrac{2x^3-x+5x^2}{x+1}=\dfrac{\left(x+1\right)\left(2x^2+3x-4\right)+4}{x+1}\)
=\(\left(2x^2+3x-4\right)+\dfrac{4}{x+1}\)
