Divide using synthetic division $$ ( 5x^{3}-27x^{2}-17x-6 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&5&-27&-17&-6\\& & 30& 18& \color{black}{6} \\ \hline &\color{blue}{5}&\color{blue}{3}&\color{blue}{1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 5x^{3}-27x^{2}-17x-6 }{ x-6 } = \color{blue}{5x^{2}+3x+1} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{6}&5&-27&-17&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 5 }&-27&-17&-6\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 5 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&5&-27&-17&-6\\& & \color{blue}{30} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 30 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}6&5&\color{orangered}{ -27 }&-17&-6\\& & \color{orangered}{30} & & \\ \hline &5&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 3 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&5&-27&-17&-6\\& & 30& \color{blue}{18} & \\ \hline &5&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 18 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}6&5&-27&\color{orangered}{ -17 }&-6\\& & 30& \color{orangered}{18} & \\ \hline &5&3&\color{orangered}{1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&5&-27&-17&-6\\& & 30& 18& \color{blue}{6} \\ \hline &5&3&\color{blue}{1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 6 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}6&5&-27&-17&\color{orangered}{ -6 }\\& & 30& 18& \color{orangered}{6} \\ \hline &\color{blue}{5}&\color{blue}{3}&\color{blue}{1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}+3x+1 } $ with a remainder of $ \color{red}{ 0 } $.