Check if $ x+2 $ is a factor of $ x^{3}-20x-8 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-2&1&0&-20&-8\\& & -2& 4& \color{black}{32} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-16}&\color{orangered}{24} \end{array} $$

Because the remainder $ \left( \color{red}{ 24 } \right) $ is not zero, we conclude that the $ x+2 $ is not a factor of $ x^{3}-20x-8$.

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 + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&-20&-8\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 1 }&0&-20&-8\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&-20&-8\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $

$$ \begin{array}{c|rrrr}-2&1&\color{orangered}{ 0 }&-20&-8\\& & \color{orangered}{-2} & & \\ \hline &1&\color{orangered}{-2}&& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&-20&-8\\& & -2& \color{blue}{4} & \\ \hline &1&\color{blue}{-2}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 4 } = \color{orangered}{ -16 } $

$$ \begin{array}{c|rrrr}-2&1&0&\color{orangered}{ -20 }&-8\\& & -2& \color{orangered}{4} & \\ \hline &1&-2&\color{orangered}{-16}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 32 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&-20&-8\\& & -2& 4& \color{blue}{32} \\ \hline &1&-2&\color{blue}{-16}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 32 } = \color{orangered}{ 24 } $

$$ \begin{array}{c|rrrr}-2&1&0&-20&\color{orangered}{ -8 }\\& & -2& 4& \color{orangered}{32} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-16}&\color{orangered}{24} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 24 }\right)$.

Synthetic Division Calculator