# Use the polynomial division algorithm to write the rational function in the form Q(x) + R(x)/D(x), where the degree of R is less than the degree of D. x^3 -5/x^2 -1

Use the polynomial division algorithm to write the rational function in the form Q(x) + R(x)/D(x), where the degree of R is less than the degree of D. ${x}^{3}-5/{x}^{2}-1$
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acilschoincg8
We use the division algorithm to find the quotient and remainder:
$\begin{array}{c}\underset{_}{\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}x\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}}\\ \left({x}^{2}-1\right)\phantom{\rule{1em}{0ex}}{x}^{3}-5\\ \underset{_}{-{x}^{3}+x}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}x-5\end{array}$
Thus
$Q\left(x\right)+\frac{R\left(x\right)}{D\left(x\right)}=x+\frac{x-5}{{x}^{2}-1}$ Substitute Q(x)=x, R(x)=x-5, and $D\left(x\right)={x}^{2}-1$
Result:
$x+\frac{x-5}{{x}^{2}-1}$