Divide $$$u^{7}$$$ by $$$u^{2} + 1$$$

Write the problem in the special format (missed terms are written with zero coefficients):

$$$\begin{array}{r|r}\hline\\u^{2}+1&u^{7}+0 u^{6}+0 u^{5}+0 u^{4}+0 u^{3}+0 u^{2}+0 u+0\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{u^{7}}{u^{2}} = u^{5}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$u^{5} \left(u^{2}+1\right) = u^{7}+u^{5}$$$.

Subtract the dividend from the obtained result: $$$\left(u^{7}\right) - \left(u^{7}+u^{5}\right) = - u^{5}$$$.

$$\begin{array}{r|rrrrrrrr:c}&{\color{SaddleBrown}u^{5}}&&&&&&&&\\\hline\\{\color{Magenta}u^{2}}+1&{\color{SaddleBrown}u^{7}}&+0 u^{6}&+0 u^{5}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{SaddleBrown}u^{7}}}{{\color{Magenta}u^{2}}} = {\color{SaddleBrown}u^{5}}\\&-\phantom{u^{7}}&&&&&&&&\\&u^{7}&+0 u^{6}&+u^{5}&&&&&&{\color{SaddleBrown}u^{5}} \left(u^{2}+1\right) = u^{7}+u^{5}\\\hline\\&&&- u^{5}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- u^{5}}{u^{2}} = - u^{3}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- u^{3} \left(u^{2}+1\right) = - u^{5}- u^{3}$$$.

Subtract the remainder from the obtained result: $$$\left(- u^{5}\right) - \left(- u^{5}- u^{3}\right) = u^{3}$$$.

$$\begin{array}{r|rrrrrrrr:c}&u^{5}&{\color{Fuchsia}- u^{3}}&&&&&&&\\\hline\\{\color{Magenta}u^{2}}+1&u^{7}&+0 u^{6}&+0 u^{5}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\\&-\phantom{u^{7}}&&&&&&&&\\&u^{7}&+0 u^{6}&+u^{5}&&&&&&\\\hline\\&&&{\color{Fuchsia}- u^{5}}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{Fuchsia}- u^{5}}}{{\color{Magenta}u^{2}}} = {\color{Fuchsia}- u^{3}}\\&&&-\phantom{- u^{5}}&&&&&&\\&&&- u^{5}&+0 u^{4}&- u^{3}&&&&{\color{Fuchsia}- u^{3}} \left(u^{2}+1\right) = - u^{5}- u^{3}\\\hline\\&&&&&u^{3}&+0 u^{2}&+0 u&+0&\end{array}$$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{u^{3}}{u^{2}} = u$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$u \left(u^{2}+1\right) = u^{3}+u$$$.

Subtract the remainder from the obtained result: $$$\left(u^{3}\right) - \left(u^{3}+u\right) = - u$$$.

$$\begin{array}{r|rrrrrrrr:c}&u^{5}&- u^{3}&{\color{Brown}+u}&&&&&&\\\hline\\{\color{Magenta}u^{2}}+1&u^{7}&+0 u^{6}&+0 u^{5}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\\&-\phantom{u^{7}}&&&&&&&&\\&u^{7}&+0 u^{6}&+u^{5}&&&&&&\\\hline\\&&&- u^{5}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\\&&&-\phantom{- u^{5}}&&&&&&\\&&&- u^{5}&+0 u^{4}&- u^{3}&&&&\\\hline\\&&&&&{\color{Brown}u^{3}}&+0 u^{2}&+0 u&+0&\frac{{\color{Brown}u^{3}}}{{\color{Magenta}u^{2}}} = {\color{Brown}u}\\&&&&&-\phantom{u^{3}}&&&&\\&&&&&u^{3}&+0 u^{2}&+u&&{\color{Brown}u} \left(u^{2}+1\right) = u^{3}+u\\\hline\\&&&&&&&- u&+0&\end{array}$$

Since the degree of the remainder is less than the degree of the divisor, we are done.

The resulting table is shown once more:

$$\begin{array}{r|rrrrrrrr:c}&{\color{SaddleBrown}u^{5}}&{\color{Fuchsia}- u^{3}}&{\color{Brown}+u}&&&&&&\text{Hints}\\\hline\\{\color{Magenta}u^{2}}+1&{\color{SaddleBrown}u^{7}}&+0 u^{6}&+0 u^{5}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{SaddleBrown}u^{7}}}{{\color{Magenta}u^{2}}} = {\color{SaddleBrown}u^{5}}\\&-\phantom{u^{7}}&&&&&&&&\\&u^{7}&+0 u^{6}&+u^{5}&&&&&&{\color{SaddleBrown}u^{5}} \left(u^{2}+1\right) = u^{7}+u^{5}\\\hline\\&&&{\color{Fuchsia}- u^{5}}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{Fuchsia}- u^{5}}}{{\color{Magenta}u^{2}}} = {\color{Fuchsia}- u^{3}}\\&&&-\phantom{- u^{5}}&&&&&&\\&&&- u^{5}&+0 u^{4}&- u^{3}&&&&{\color{Fuchsia}- u^{3}} \left(u^{2}+1\right) = - u^{5}- u^{3}\\\hline\\&&&&&{\color{Brown}u^{3}}&+0 u^{2}&+0 u&+0&\frac{{\color{Brown}u^{3}}}{{\color{Magenta}u^{2}}} = {\color{Brown}u}\\&&&&&-\phantom{u^{3}}&&&&\\&&&&&u^{3}&+0 u^{2}&+u&&{\color{Brown}u} \left(u^{2}+1\right) = u^{3}+u\\\hline\\&&&&&&&- u&+0&\end{array}$$

Therefore, $$$\frac{u^{7}}{u^{2} + 1} = \left(u^{5} - u^{3} + u\right) + \frac{- u}{u^{2} + 1}$$$.