1
Solved example of simplification of algebraic fractions
$\frac{x^4-\frac{4}{3} x^3-2x^2-\frac{4}{3}\cdot x+1}{x-2}$
$\frac{x^4-1\cdot \frac{4}{3}x^3-2x^2-x\frac{4}{3}+1}{x-2}$
3
Multiplying the fraction and term
$\frac{x^4-\frac{4}{3}x^3-2x^2-\frac{4}{3}x+1}{x-2}$
4
Divide $x^4-\frac{4}{3}x^3-2x^2-\frac{4}{3}x+1$ by $x-2$
$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}x^{3}+2x^{2}+2x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}-2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-x^{4}+2x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}+2x^{3};}\phantom{;}2x^{3}-2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-2x^{3}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-2x^{3}+4x^{2}-;x^n;}\phantom{;}2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{-2x^{2}+4x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-2x^{2}+4x\phantom{;}-;x^n-;x^n;}\phantom{;}4x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n;}\underline{-4x\phantom{;}+8\phantom{;}\phantom{;}}\\\phantom{;;;-4x\phantom{;}+8\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}9\phantom{;}\phantom{;}\\\end{array}$
$x^{3}+2x^{2}+2x+4+\frac{9}{x-2}$
Answer
$x^{3}+2x^{2}+2x+4+\frac{9}{x-2}$