Divide $$$u^{3}$$$ by $$$u - 1$$$

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

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

Step 1

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

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

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

Subtract the dividend from the obtained result: $$$\left(u^{3}\right) - \left(u^{3}- u^{2}\right) = u^{2}$$$.

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

Step 2

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

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

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

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

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

Step 3

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

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

Multiply it by the divisor: $$$1 \left(u-1\right) = u-1$$$.

Subtract the remainder from the obtained result: $$$\left(u\right) - \left(u-1\right) = 1$$$.

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

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

The resulting table is shown once more:

$$\begin{array}{r|rrrr:c}&{\color{BlueViolet}u^{2}}&{\color{SaddleBrown}+u}&{\color{Green}+1}&&\text{Hints}\\\hline\\{\color{Magenta}u}-1&{\color{BlueViolet}u^{3}}&+0 u^{2}&+0 u&+0&\frac{{\color{BlueViolet}u^{3}}}{{\color{Magenta}u}} = {\color{BlueViolet}u^{2}}\\&-\phantom{u^{3}}&&&&\\&u^{3}&- u^{2}&&&{\color{BlueViolet}u^{2}} \left(u-1\right) = u^{3}- u^{2}\\\hline\\&&{\color{SaddleBrown}u^{2}}&+0 u&+0&\frac{{\color{SaddleBrown}u^{2}}}{{\color{Magenta}u}} = {\color{SaddleBrown}u}\\&&-\phantom{u^{2}}&&&\\&&u^{2}&- u&&{\color{SaddleBrown}u} \left(u-1\right) = u^{2}- u\\\hline\\&&&{\color{Green}u}&+0&\frac{{\color{Green}u}}{{\color{Magenta}u}} = {\color{Green}1}\\&&&-\phantom{u}&&\\&&&u&-1&{\color{Green}1} \left(u-1\right) = u-1\\\hline\\&&&&1&\end{array}$$

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