Divide using synthetic division $$ ( x^{2}-2x-63 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}9&1&-2&-63\\& & 9& \color{black}{63} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{2}-2x-63 }{ x-9 } = \color{blue}{x+7} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{9}&1&-2&-63\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}9&\color{orangered}{ 1 }&-2&-63\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrr}\color{blue}{9}&1&-2&-63\\& & \color{blue}{9} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 9 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrr}9&1&\color{orangered}{ -2 }&-63\\& & \color{orangered}{9} & \\ \hline &1&\color{orangered}{7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 7 } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrr}\color{blue}{9}&1&-2&-63\\& & 9& \color{blue}{63} \\ \hline &1&\color{blue}{7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -63 } + \color{orangered}{ 63 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}9&1&-2&\color{orangered}{ -63 }\\& & 9& \color{orangered}{63} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x+7 } $ with a remainder of $ \color{red}{ 0 } $.