# Continued fractions with rational functions

Express the following rational function in continued-fraction form: $${4x^2+3x-7\over 2x^3+x^2-x+5}$$ The answer is : $${4 \over 2x- \frac{1}{2}} + { \frac{23}{8} \over x-\frac{63}{92}}-{\frac{406}{529} \over x+\frac{33}{23}}\tag{inline continued fraction}$$ which means $$\cfrac{4}{2x- \frac{1}{2}+\cfrac{\frac{23}{8}}{x-\frac{63}{92}-\cfrac{\frac{406}{529}}{x+\frac{33}{23}}}}$$

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Here is the solution using integer coefficients \begin{align} \cfrac{4x^2+3x-7}{2x^3+x^2-x+5} &=\cfrac1{\cfrac{2x^3+x^2-x+5}{4x^2+3x-7}}\\ &=\cfrac{8}{4x-1+\cfrac{23x+33}{4x^2+3x-7}}\\ &=\cfrac{8}{4x-1+\cfrac{529}{92x-63-\cfrac{1624}{23x+33}}}\\ \end{align} The book's answer can be shown to be equal by cancelling fractions.