Divide $$$u^{6}$$$ by $$$u^{2} + 1$$$

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

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

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{u^{6}}{u^{2}} = u^{4}$$$.

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

Multiply it by the divisor: $$$u^{4} \left(u^{2}+1\right) = u^{6}+u^{4}$$$.

Subtract the dividend from the obtained result: $$$\left(u^{6}\right) - \left(u^{6}+u^{4}\right) = - u^{4}$$$.

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

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- u^{4}}{u^{2}} = - u^{2}$$$.

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

Multiply it by the divisor: $$$- u^{2} \left(u^{2}+1\right) = - u^{4}- u^{2}$$$.

Subtract the remainder from the obtained result: $$$\left(- u^{4}\right) - \left(- u^{4}- u^{2}\right) = u^{2}$$$.

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

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{u^{2}}{u^{2}} = 1$$$.

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

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

Subtract the remainder from the obtained result: $$$\left(u^{2}\right) - \left(u^{2}+1\right) = -1$$$.

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

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

The resulting table is shown once more:

$$\begin{array}{r|rrrrrrr:c}&{\color{DarkMagenta}u^{4}}&{\color{OrangeRed}- u^{2}}&{\color{DeepPink}+1}&&&&&\text{Hints}\\\hline\\{\color{Magenta}u^{2}}+1&{\color{DarkMagenta}u^{6}}&+0 u^{5}&+0 u^{4}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{DarkMagenta}u^{6}}}{{\color{Magenta}u^{2}}} = {\color{DarkMagenta}u^{4}}\\&-\phantom{u^{6}}&&&&&&&\\&u^{6}&+0 u^{5}&+u^{4}&&&&&{\color{DarkMagenta}u^{4}} \left(u^{2}+1\right) = u^{6}+u^{4}\\\hline\\&&&{\color{OrangeRed}- u^{4}}&+0 u^{3}&+0 u^{2}&+0 u&+0&\frac{{\color{OrangeRed}- u^{4}}}{{\color{Magenta}u^{2}}} = {\color{OrangeRed}- u^{2}}\\&&&-\phantom{- u^{4}}&&&&&\\&&&- u^{4}&+0 u^{3}&- u^{2}&&&{\color{OrangeRed}- u^{2}} \left(u^{2}+1\right) = - u^{4}- u^{2}\\\hline\\&&&&&{\color{DeepPink}u^{2}}&+0 u&+0&\frac{{\color{DeepPink}u^{2}}}{{\color{Magenta}u^{2}}} = {\color{DeepPink}1}\\&&&&&-\phantom{u^{2}}&&&\\&&&&&u^{2}&+0 u&+1&{\color{DeepPink}1} \left(u^{2}+1\right) = u^{2}+1\\\hline\\&&&&&&&-1&\end{array}$$

Therefore, $$$\frac{u^{6}}{u^{2} + 1} = \left(u^{4} - u^{2} + 1\right) + \frac{-1}{u^{2} + 1}$$$.