Divide using synthetic division $$ ( x^{2}-x-29 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}6&1&-1&-29\\& & 6& \color{black}{30} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{orangered}{1} \end{array} $$The solution is:
$$ \frac{ x^{2}-x-29 }{ x-6 } = \color{blue}{x+5} ~+~ \frac{ \color{red}{ 1 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{6}&1&-1&-29\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}6&\color{orangered}{ 1 }&-1&-29\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&1&-1&-29\\& & \color{blue}{6} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 6 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}6&1&\color{orangered}{ -1 }&-29\\& & \color{orangered}{6} & \\ \hline &1&\color{orangered}{5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 5 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&1&-1&-29\\& & 6& \color{blue}{30} \\ \hline &1&\color{blue}{5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 30 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrr}6&1&-1&\color{orangered}{ -29 }\\& & 6& \color{orangered}{30} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{orangered}{1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x+5 } $ with a remainder of $ \color{red}{ 1 } $.