Divide using synthetic division $$ ( x^{3}+17x^{2}+70x-54 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&1&17&70&-54\\& & 6& 138& \color{black}{1248} \\ \hline &\color{blue}{1}&\color{blue}{23}&\color{blue}{208}&\color{orangered}{1194} \end{array} $$The solution is:
$$ \frac{ x^{3}+17x^{2}+70x-54 }{ x-6 } = \color{blue}{x^{2}+23x+208} ~+~ \frac{ \color{red}{ 1194 } }{ x-6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&17&70&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 1 }&17&70&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&17&70&-54\\& & \color{blue}{6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 6 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrr}6&1&\color{orangered}{ 17 }&70&-54\\& & \color{orangered}{6} & & \\ \hline &1&\color{orangered}{23}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 23 } = \color{blue}{ 138 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&17&70&-54\\& & 6& \color{blue}{138} & \\ \hline &1&\color{blue}{23}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ 138 } = \color{orangered}{ 208 } $
$$ \begin{array}{c|rrrr}6&1&17&\color{orangered}{ 70 }&-54\\& & 6& \color{orangered}{138} & \\ \hline &1&23&\color{orangered}{208}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 208 } = \color{blue}{ 1248 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&17&70&-54\\& & 6& 138& \color{blue}{1248} \\ \hline &1&23&\color{blue}{208}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 1248 } = \color{orangered}{ 1194 } $
$$ \begin{array}{c|rrrr}6&1&17&70&\color{orangered}{ -54 }\\& & 6& 138& \color{orangered}{1248} \\ \hline &\color{blue}{1}&\color{blue}{23}&\color{blue}{208}&\color{orangered}{1194} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+23x+208 } $ with a remainder of $ \color{red}{ 1194 } $.