Check if $ x-6 $ is a factor of $ -2x^{4}+11x^{3}+6x^{2}+x+3 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}6&-2&11&6&1&3\\& & -12& -6& 0& \color{black}{6} \\ \hline &\color{blue}{-2}&\color{blue}{-1}&\color{blue}{0}&\color{blue}{1}&\color{orangered}{9} \end{array} $$

Because the remainder $ \left( \color{red}{ 9 } \right) $ is not zero, we conclude that the $ x-6 $ is not a factor of $ -2x^{4}+11x^{3}+6x^{2}+x+3$.

First we need to create a synthetic division table.

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.

$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&11&6&1&3\\& & & & & \\ \hline &&&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrrr}6&\color{orangered}{ -2 }&11&6&1&3\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -12 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&11&6&1&3\\& & \color{blue}{-12} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -1 } $

$$ \begin{array}{c|rrrrr}6&-2&\color{orangered}{ 11 }&6&1&3\\& & \color{orangered}{-12} & & & \\ \hline &-2&\color{orangered}{-1}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -6 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&11&6&1&3\\& & -12& \color{blue}{-6} & & \\ \hline &-2&\color{blue}{-1}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}6&-2&11&\color{orangered}{ 6 }&1&3\\& & -12& \color{orangered}{-6} & & \\ \hline &-2&-1&\color{orangered}{0}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&11&6&1&3\\& & -12& -6& \color{blue}{0} & \\ \hline &-2&-1&\color{blue}{0}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $

$$ \begin{array}{c|rrrrr}6&-2&11&6&\color{orangered}{ 1 }&3\\& & -12& -6& \color{orangered}{0} & \\ \hline &-2&-1&0&\color{orangered}{1}& \end{array} $$

Step 8 : 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 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&11&6&1&3\\& & -12& -6& 0& \color{blue}{6} \\ \hline &-2&-1&0&\color{blue}{1}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 6 } = \color{orangered}{ 9 } $

$$ \begin{array}{c|rrrrr}6&-2&11&6&1&\color{orangered}{ 3 }\\& & -12& -6& 0& \color{orangered}{6} \\ \hline &\color{blue}{-2}&\color{blue}{-1}&\color{blue}{0}&\color{blue}{1}&\color{orangered}{9} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 9 }\right)$.

Synthetic Division Calculator