Divide $$$x^{2} - 7 x + 10$$$ by $$$x - 5$$$

Write the problem in the special format:

$$$\begin{array}{r|r}\hline\\x-5&x^{2}- 7 x+10\end{array}$$$

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: $$$x \left(x-5\right) = x^{2}- 5 x$$$.

Subtract the dividend from the obtained result: $$$\left(x^{2}- 7 x+10\right) - \left(x^{2}- 5 x\right) = - 2 x+10$$$.

$$\begin{array}{r|rrr:c}&{\color{Green}x}&&&\\\hline\\{\color{Magenta}x}-5&{\color{Green}x^{2}}&- 7 x&+10&\frac{{\color{Green}x^{2}}}{{\color{Magenta}x}} = {\color{Green}x}\\&-\phantom{x^{2}}&&&\\&x^{2}&- 5 x&&{\color{Green}x} \left(x-5\right) = x^{2}- 5 x\\\hline\\&&- 2 x&+10&\end{array}$$

Step 2

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

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

Multiply it by the divisor: $$$- 2 \left(x-5\right) = - 2 x+10$$$.

Subtract the remainder from the obtained result: $$$\left(- 2 x+10\right) - \left(- 2 x+10\right) = $$$.

$$\begin{array}{r|rrr:c}&x&{\color{SaddleBrown}-2}&&\\\hline\\{\color{Magenta}x}-5&x^{2}&- 7 x&+10&\\&-\phantom{x^{2}}&&&\\&x^{2}&- 5 x&&\\\hline\\&&{\color{SaddleBrown}- 2 x}&+10&\frac{{\color{SaddleBrown}- 2 x}}{{\color{Magenta}x}} = {\color{SaddleBrown}-2}\\&&-\phantom{- 2 x}&&\\&&- 2 x&+10&{\color{SaddleBrown}-2} \left(x-5\right) = - 2 x+10\\\hline\\&&&0&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrr:c}&{\color{Green}x}&{\color{SaddleBrown}-2}&&\text{Hints}\\\hline\\{\color{Magenta}x}-5&{\color{Green}x^{2}}&- 7 x&+10&\frac{{\color{Green}x^{2}}}{{\color{Magenta}x}} = {\color{Green}x}\\&-\phantom{x^{2}}&&&\\&x^{2}&- 5 x&&{\color{Green}x} \left(x-5\right) = x^{2}- 5 x\\\hline\\&&{\color{SaddleBrown}- 2 x}&+10&\frac{{\color{SaddleBrown}- 2 x}}{{\color{Magenta}x}} = {\color{SaddleBrown}-2}\\&&-\phantom{- 2 x}&&\\&&- 2 x&+10&{\color{SaddleBrown}-2} \left(x-5\right) = - 2 x+10\\\hline\\&&&0&\end{array}$$

Therefore, $$$\frac{x^{2} - 7 x + 10}{x - 5} = \left(x - 2\right) + \frac{0}{x - 5} = x - 2$$$.