Check if $ x+6 $ is a factor of $ 2x^{3}+11x^{2}-5x-3 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-6&2&11&-5&-3\\& & -12& 6& \color{black}{-6} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{1}&\color{orangered}{-9} \end{array} $$

Because the remainder $ \left( \color{red}{ -9 } \right) $ is not zero, we conclude that the $ x+6 $ is not a factor of $ 2x^{3}+11x^{2}-5x-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.

$$ \begin{array}{c|rrrr}\color{blue}{-6}&2&11&-5&-3\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 2 }&11&-5&-3\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{-6}&2&11&-5&-3\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{2}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-6&2&\color{orangered}{ 11 }&-5&-3\\& & \color{orangered}{-12} & & \\ \hline &2&\color{orangered}{-1}&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-6}&2&11&-5&-3\\& & -12& \color{blue}{6} & \\ \hline &2&\color{blue}{-1}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 6 } = \color{orangered}{ 1 } $

$$ \begin{array}{c|rrrr}-6&2&11&\color{orangered}{ -5 }&-3\\& & -12& \color{orangered}{6} & \\ \hline &2&-1&\color{orangered}{1}& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-6}&2&11&-5&-3\\& & -12& 6& \color{blue}{-6} \\ \hline &2&-1&\color{blue}{1}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $

$$ \begin{array}{c|rrrr}-6&2&11&-5&\color{orangered}{ -3 }\\& & -12& 6& \color{orangered}{-6} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{1}&\color{orangered}{-9} \end{array} $$

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

Synthetic Division Calculator