Divide $$$x^{2}$$$ by $$$x - 7$$$

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

$$$\begin{array}{r|r}\hline\\x-7&x^{2}+0 x+0\end{array}$$$

Step 1

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

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

Multiply it by the divisor: $$$x \left(x-7\right) = x^{2}- 7 x$$$.

Subtract the dividend from the obtained result: $$$\left(x^{2}\right) - \left(x^{2}- 7 x\right) = 7 x$$$.

$$\begin{array}{r|rrr:c}&{\color{Fuchsia}x}&&&\\\hline\\{\color{Magenta}x}-7&{\color{Fuchsia}x^{2}}&+0 x&+0&\frac{{\color{Fuchsia}x^{2}}}{{\color{Magenta}x}} = {\color{Fuchsia}x}\\&-\phantom{x^{2}}&&&\\&x^{2}&- 7 x&&{\color{Fuchsia}x} \left(x-7\right) = x^{2}- 7 x\\\hline\\&&7 x&+0&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{7 x}{x} = 7$$$.

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

Multiply it by the divisor: $$$7 \left(x-7\right) = 7 x-49$$$.

Subtract the remainder from the obtained result: $$$\left(7 x\right) - \left(7 x-49\right) = 49$$$.

$$\begin{array}{r|rrr:c}&x&{\color{Chartreuse}+7}&&\\\hline\\{\color{Magenta}x}-7&x^{2}&+0 x&+0&\\&-\phantom{x^{2}}&&&\\&x^{2}&- 7 x&&\\\hline\\&&{\color{Chartreuse}7 x}&+0&\frac{{\color{Chartreuse}7 x}}{{\color{Magenta}x}} = {\color{Chartreuse}7}\\&&-\phantom{7 x}&&\\&&7 x&-49&{\color{Chartreuse}7} \left(x-7\right) = 7 x-49\\\hline\\&&&49&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrr:c}&{\color{Fuchsia}x}&{\color{Chartreuse}+7}&&\text{Hints}\\\hline\\{\color{Magenta}x}-7&{\color{Fuchsia}x^{2}}&+0 x&+0&\frac{{\color{Fuchsia}x^{2}}}{{\color{Magenta}x}} = {\color{Fuchsia}x}\\&-\phantom{x^{2}}&&&\\&x^{2}&- 7 x&&{\color{Fuchsia}x} \left(x-7\right) = x^{2}- 7 x\\\hline\\&&{\color{Chartreuse}7 x}&+0&\frac{{\color{Chartreuse}7 x}}{{\color{Magenta}x}} = {\color{Chartreuse}7}\\&&-\phantom{7 x}&&\\&&7 x&-49&{\color{Chartreuse}7} \left(x-7\right) = 7 x-49\\\hline\\&&&49&\end{array}$$

Therefore, $$$\frac{x^{2}}{x - 7} = \left(x + 7\right) + \frac{49}{x - 7}$$$.