Check if $ x+1 $ is a factor of $ 12x^{3}+29x^{2}+8x-4 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-1&12&29&8&-4\\& & -12& -17& \color{black}{9} \\ \hline &\color{blue}{12}&\color{blue}{17}&\color{blue}{-9}&\color{orangered}{5} \end{array} $$

Because the remainder $ \left( \color{red}{ 5 } \right) $ is not zero, we conclude that the $ x+1 $ is not a factor of $ 12x^{3}+29x^{2}+8x-4$.

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 + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&29&8&-4\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 12 }&29&8&-4\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 12 } = \color{blue}{ -12 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&29&8&-4\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{12}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 17 } $

$$ \begin{array}{c|rrrr}-1&12&\color{orangered}{ 29 }&8&-4\\& & \color{orangered}{-12} & & \\ \hline &12&\color{orangered}{17}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 17 } = \color{blue}{ -17 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&29&8&-4\\& & -12& \color{blue}{-17} & \\ \hline &12&\color{blue}{17}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -17 \right) } = \color{orangered}{ -9 } $

$$ \begin{array}{c|rrrr}-1&12&29&\color{orangered}{ 8 }&-4\\& & -12& \color{orangered}{-17} & \\ \hline &12&17&\color{orangered}{-9}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&29&8&-4\\& & -12& -17& \color{blue}{9} \\ \hline &12&17&\color{blue}{-9}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 9 } = \color{orangered}{ 5 } $

$$ \begin{array}{c|rrrr}-1&12&29&8&\color{orangered}{ -4 }\\& & -12& -17& \color{orangered}{9} \\ \hline &\color{blue}{12}&\color{blue}{17}&\color{blue}{-9}&\color{orangered}{5} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 5 }\right)$.

Synthetic Division Calculator