Divide using synthetic division $$ ( x^{3}-3x^{2}+20 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&-3&0&20\\& & -6& 54& \color{black}{-324} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{54}&\color{orangered}{-304} \end{array} $$The solution is:
$$ \frac{ x^{3}-3x^{2}+20 }{ x+6 } = \color{blue}{x^{2}-9x+54} \color{red}{~-~} \frac{ \color{red}{ 304 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-3&0&20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&-3&0&20\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-3&0&20\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ -3 }&0&20\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-3&0&20\\& & -6& \color{blue}{54} & \\ \hline &1&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 54 } = \color{orangered}{ 54 } $
$$ \begin{array}{c|rrrr}-6&1&-3&\color{orangered}{ 0 }&20\\& & -6& \color{orangered}{54} & \\ \hline &1&-9&\color{orangered}{54}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 54 } = \color{blue}{ -324 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-3&0&20\\& & -6& 54& \color{blue}{-324} \\ \hline &1&-9&\color{blue}{54}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -324 \right) } = \color{orangered}{ -304 } $
$$ \begin{array}{c|rrrr}-6&1&-3&0&\color{orangered}{ 20 }\\& & -6& 54& \color{orangered}{-324} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{54}&\color{orangered}{-304} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-9x+54 } $ with a remainder of $ \color{red}{ -304 } $.