Final answer to the problem
$\frac{3}{2}x^{3}+\frac{3}{4}x^{2}-\frac{17}{8}x-\frac{17}{16}+\frac{\frac{303}{16}}{2x-1}$
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Step-by-step Solution
Specify the solving method
1
Divide $3x^4-5x^2+20$ by $2x-1$
$\begin{array}{l}\phantom{\phantom{;}2x\phantom{;}-1;}{\phantom{;}\frac{3}{2}x^{3}+\frac{3}{4}x^{2}-\frac{17}{8}x\phantom{;}-\frac{17}{16}\phantom{;}\phantom{;}}\\\phantom{;}2x\phantom{;}-1\overline{\smash{)}\phantom{;}3x^{4}\phantom{-;x^n}-5x^{2}\phantom{-;x^n}+20\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2x\phantom{;}-1;}\underline{-3x^{4}+\frac{3}{2}x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{4}+\frac{3}{2}x^{3};}\phantom{;}\frac{3}{2}x^{3}-5x^{2}\phantom{-;x^n}+20\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n;}\underline{-\frac{3}{2}x^{3}+\frac{3}{4}x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-\frac{3}{2}x^{3}+\frac{3}{4}x^{2}-;x^n;}-\frac{17}{4}x^{2}\phantom{-;x^n}+20\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n;}\underline{\phantom{;}\frac{17}{4}x^{2}-\frac{17}{8}x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}\frac{17}{4}x^{2}-\frac{17}{8}x\phantom{;}-;x^n-;x^n;}-\frac{17}{8}x\phantom{;}+20\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n;}\underline{\phantom{;}\frac{17}{8}x\phantom{;}-\frac{17}{16}\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}\frac{17}{8}x\phantom{;}-\frac{17}{16}\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}\frac{303}{16}\phantom{;}\phantom{;}\\\end{array}$
2
Resulting polynomial
$\frac{3}{2}x^{3}+\frac{3}{4}x^{2}-\frac{17}{8}x-\frac{17}{16}+\frac{\frac{303}{16}}{2x-1}$
Final answer to the problem
$\frac{3}{2}x^{3}+\frac{3}{4}x^{2}-\frac{17}{8}x-\frac{17}{16}+\frac{\frac{303}{16}}{2x-1}$