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

The synthetic division table is:

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

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

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.

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

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

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

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

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

Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -3 } $

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

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 3 } $.

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

Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 3 } = \color{orangered}{ 1 } $

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

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

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

Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -3 } $

$$ \begin{array}{c|rrrr}-1&2&-1&-2&\color{orangered}{ -2 }\\& & -2& 3& \color{orangered}{-1} \\ \hline &\color{blue}{2}&\color{blue}{-3}&\color{blue}{1}&\color{orangered}{-3} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -3 }\right)$.

Synthetic Division Calculator