Check if $ x+4 $ is a factor of $ x^{3}+14x^{2}+36x-16 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-4&1&14&36&-16\\& & -4& -40& \color{black}{16} \\ \hline &\color{blue}{1}&\color{blue}{10}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$

Because the remainder equals zero, we conclude that the $ x+4 $ is a factor of the $ x^{3}+14x^{2}+36x-16 $.

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 + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&14&36&-16\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 1 }&14&36&-16\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&14&36&-16\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 10 } $

$$ \begin{array}{c|rrrr}-4&1&\color{orangered}{ 14 }&36&-16\\& & \color{orangered}{-4} & & \\ \hline &1&\color{orangered}{10}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 10 } = \color{blue}{ -40 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&14&36&-16\\& & -4& \color{blue}{-40} & \\ \hline &1&\color{blue}{10}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -4 } $

$$ \begin{array}{c|rrrr}-4&1&14&\color{orangered}{ 36 }&-16\\& & -4& \color{orangered}{-40} & \\ \hline &1&10&\color{orangered}{-4}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&14&36&-16\\& & -4& -40& \color{blue}{16} \\ \hline &1&10&\color{blue}{-4}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 16 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrr}-4&1&14&36&\color{orangered}{ -16 }\\& & -4& -40& \color{orangered}{16} \\ \hline &\color{blue}{1}&\color{blue}{10}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.

Synthetic Division Calculator