Find remainder when $ x^{3}+15x^{2}+68x+84 $ is divided by $ x+6 $.
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&15&68&84\\& & -6& -54& \color{black}{-84} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{14}&\color{orangered}{0} \end{array} $$The remainder when $ x^{3}+15x^{2}+68x+84 $ is divided by $ x+6 $ is $ \, \color{red}{ 0 } $.
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&15&68&84\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&15&68&84\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&15&68&84\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ 15 }&68&84\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 9 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&15&68&84\\& & -6& \color{blue}{-54} & \\ \hline &1&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 68 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}-6&1&15&\color{orangered}{ 68 }&84\\& & -6& \color{orangered}{-54} & \\ \hline &1&9&\color{orangered}{14}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 14 } = \color{blue}{ -84 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&15&68&84\\& & -6& -54& \color{blue}{-84} \\ \hline &1&9&\color{blue}{14}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 84 } + \color{orangered}{ \left( -84 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-6&1&15&68&\color{orangered}{ 84 }\\& & -6& -54& \color{orangered}{-84} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{14}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.