Divide using synthetic division $$ ( x^{2}-2x+7 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&1&-2&7\\& & 5& \color{black}{15} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{orangered}{22} \end{array} $$The solution is:
$$ \frac{ x^{2}-2x+7 }{ x-5 } = \color{blue}{x+3} ~+~ \frac{ \color{red}{ 22 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{5}&1&-2&7\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ 1 }&-2&7\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&1&-2&7\\& & \color{blue}{5} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 5 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrr}5&1&\color{orangered}{ -2 }&7\\& & \color{orangered}{5} & \\ \hline &1&\color{orangered}{3}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 3 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&1&-2&7\\& & 5& \color{blue}{15} \\ \hline &1&\color{blue}{3}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 15 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrr}5&1&-2&\color{orangered}{ 7 }\\& & 5& \color{orangered}{15} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{orangered}{22} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x+3 } $ with a remainder of $ \color{red}{ 22 } $.