Divide $$$v^{3}$$$ by $$$v^{2} + 1$$$
Related calculators: Synthetic Division Calculator, Long Division Calculator
Your Input
Find $$$\frac{v^{3}}{v^{2} + 1}$$$ using long division.
Solution
Write the problem in the special format (missed terms are written with zero coefficients):
$$$\begin{array}{r|r}\hline\\v^{2}+1&v^{3}+0 v^{2}+0 v+0\end{array}$$$
Step 1
Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{v^{3}}{v^{2}} = v$$$.
Write down the calculated result in the upper part of the table.
Multiply it by the divisor: $$$v \left(v^{2}+1\right) = v^{3}+v$$$.
Subtract the dividend from the obtained result: $$$\left(v^{3}\right) - \left(v^{3}+v\right) = - v$$$.
$$\begin{array}{r|rrrr:c}&{\color{Blue}v}&&&&\\\hline\\{\color{Magenta}v^{2}}+1&{\color{Blue}v^{3}}&+0 v^{2}&+0 v&+0&\frac{{\color{Blue}v^{3}}}{{\color{Magenta}v^{2}}} = {\color{Blue}v}\\&-\phantom{v^{3}}&&&&\\&v^{3}&+0 v^{2}&+v&&{\color{Blue}v} \left(v^{2}+1\right) = v^{3}+v\\\hline\\&&&- v&+0&\end{array}$$Since the degree of the remainder is less than the degree of the divisor, we are done.
The resulting table is shown once more:
$$\begin{array}{r|rrrr:c}&{\color{Blue}v}&&&&\text{Hints}\\\hline\\{\color{Magenta}v^{2}}+1&{\color{Blue}v^{3}}&+0 v^{2}&+0 v&+0&\frac{{\color{Blue}v^{3}}}{{\color{Magenta}v^{2}}} = {\color{Blue}v}\\&-\phantom{v^{3}}&&&&\\&v^{3}&+0 v^{2}&+v&&{\color{Blue}v} \left(v^{2}+1\right) = v^{3}+v\\\hline\\&&&- v&+0&\end{array}$$Therefore, $$$\frac{v^{3}}{v^{2} + 1} = v + \frac{- v}{v^{2} + 1}$$$.
Answer
$$$\frac{v^{3}}{v^{2} + 1} = v + \frac{- v}{v^{2} + 1}$$$A