Divide $$$x^{4}$$$ by $$$x - 1$$$

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

$$$\begin{array}{r|r}\hline\\x-1&x^{4}+0 x^{3}+0 x^{2}+0 x+0\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{x^{4}}{x} = x^{3}$$$.

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

Multiply it by the divisor: $$$x^{3} \left(x-1\right) = x^{4}- x^{3}$$$.

Subtract the dividend from the obtained result: $$$\left(x^{4}\right) - \left(x^{4}- x^{3}\right) = x^{3}$$$.

$$\begin{array}{r|rrrrr:c}&{\color{BlueViolet}x^{3}}&&&&&\\\hline\\{\color{Magenta}x}-1&{\color{BlueViolet}x^{4}}&+0 x^{3}&+0 x^{2}&+0 x&+0&\frac{{\color{BlueViolet}x^{4}}}{{\color{Magenta}x}} = {\color{BlueViolet}x^{3}}\\&-\phantom{x^{4}}&&&&&\\&x^{4}&- x^{3}&&&&{\color{BlueViolet}x^{3}} \left(x-1\right) = x^{4}- x^{3}\\\hline\\&&x^{3}&+0 x^{2}&+0 x&+0&\end{array}$$

Step 2

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

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

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

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

$$\begin{array}{r|rrrrr:c}&x^{3}&{\color{GoldenRod}+x^{2}}&&&&\\\hline\\{\color{Magenta}x}-1&x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\\&-\phantom{x^{4}}&&&&&\\&x^{4}&- x^{3}&&&&\\\hline\\&&{\color{GoldenRod}x^{3}}&+0 x^{2}&+0 x&+0&\frac{{\color{GoldenRod}x^{3}}}{{\color{Magenta}x}} = {\color{GoldenRod}x^{2}}\\&&-\phantom{x^{3}}&&&&\\&&x^{3}&- x^{2}&&&{\color{GoldenRod}x^{2}} \left(x-1\right) = x^{3}- x^{2}\\\hline\\&&&x^{2}&+0 x&+0&\end{array}$$

Step 3

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

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

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

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

$$\begin{array}{r|rrrrr:c}&x^{3}&+x^{2}&{\color{DarkBlue}+x}&&&\\\hline\\{\color{Magenta}x}-1&x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\\&-\phantom{x^{4}}&&&&&\\&x^{4}&- x^{3}&&&&\\\hline\\&&x^{3}&+0 x^{2}&+0 x&+0&\\&&-\phantom{x^{3}}&&&&\\&&x^{3}&- x^{2}&&&\\\hline\\&&&{\color{DarkBlue}x^{2}}&+0 x&+0&\frac{{\color{DarkBlue}x^{2}}}{{\color{Magenta}x}} = {\color{DarkBlue}x}\\&&&-\phantom{x^{2}}&&&\\&&&x^{2}&- x&&{\color{DarkBlue}x} \left(x-1\right) = x^{2}- x\\\hline\\&&&&x&+0&\end{array}$$

Step 4

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

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

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

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

$$\begin{array}{r|rrrrr:c}&x^{3}&+x^{2}&+x&{\color{Violet}+1}&&\\\hline\\{\color{Magenta}x}-1&x^{4}&+0 x^{3}&+0 x^{2}&+0 x&+0&\\&-\phantom{x^{4}}&&&&&\\&x^{4}&- x^{3}&&&&\\\hline\\&&x^{3}&+0 x^{2}&+0 x&+0&\\&&-\phantom{x^{3}}&&&&\\&&x^{3}&- x^{2}&&&\\\hline\\&&&x^{2}&+0 x&+0&\\&&&-\phantom{x^{2}}&&&\\&&&x^{2}&- x&&\\\hline\\&&&&{\color{Violet}x}&+0&\frac{{\color{Violet}x}}{{\color{Magenta}x}} = {\color{Violet}1}\\&&&&-\phantom{x}&&\\&&&&x&-1&{\color{Violet}1} \left(x-1\right) = x-1\\\hline\\&&&&&1&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrrrr:c}&{\color{BlueViolet}x^{3}}&{\color{GoldenRod}+x^{2}}&{\color{DarkBlue}+x}&{\color{Violet}+1}&&\text{Hints}\\\hline\\{\color{Magenta}x}-1&{\color{BlueViolet}x^{4}}&+0 x^{3}&+0 x^{2}&+0 x&+0&\frac{{\color{BlueViolet}x^{4}}}{{\color{Magenta}x}} = {\color{BlueViolet}x^{3}}\\&-\phantom{x^{4}}&&&&&\\&x^{4}&- x^{3}&&&&{\color{BlueViolet}x^{3}} \left(x-1\right) = x^{4}- x^{3}\\\hline\\&&{\color{GoldenRod}x^{3}}&+0 x^{2}&+0 x&+0&\frac{{\color{GoldenRod}x^{3}}}{{\color{Magenta}x}} = {\color{GoldenRod}x^{2}}\\&&-\phantom{x^{3}}&&&&\\&&x^{3}&- x^{2}&&&{\color{GoldenRod}x^{2}} \left(x-1\right) = x^{3}- x^{2}\\\hline\\&&&{\color{DarkBlue}x^{2}}&+0 x&+0&\frac{{\color{DarkBlue}x^{2}}}{{\color{Magenta}x}} = {\color{DarkBlue}x}\\&&&-\phantom{x^{2}}&&&\\&&&x^{2}&- x&&{\color{DarkBlue}x} \left(x-1\right) = x^{2}- x\\\hline\\&&&&{\color{Violet}x}&+0&\frac{{\color{Violet}x}}{{\color{Magenta}x}} = {\color{Violet}1}\\&&&&-\phantom{x}&&\\&&&&x&-1&{\color{Violet}1} \left(x-1\right) = x-1\\\hline\\&&&&&1&\end{array}$$

Therefore, $$$\frac{x^{4}}{x - 1} = \left(x^{3} + x^{2} + x + 1\right) + \frac{1}{x - 1}$$$.