Check if $ x+1 $ is a factor of $ -3x^{3}+8x^{2}+3x-7 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-1&-3&8&3&-7\\& & 3& -11& \color{black}{8} \\ \hline &\color{blue}{-3}&\color{blue}{11}&\color{blue}{-8}&\color{orangered}{1} \end{array} $$

Because the remainder $ \left( \color{red}{ 1 } \right) $ is not zero, we conclude that the $ x+1 $ is not a factor of $ -3x^{3}+8x^{2}+3x-7$.

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.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&-3&8&3&-7\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-1&\color{orangered}{ -3 }&8&3&-7\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 3 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&-3&8&3&-7\\& & \color{blue}{3} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 3 } = \color{orangered}{ 11 } $

$$ \begin{array}{c|rrrr}-1&-3&\color{orangered}{ 8 }&3&-7\\& & \color{orangered}{3} & & \\ \hline &-3&\color{orangered}{11}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 11 } = \color{blue}{ -11 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&-3&8&3&-7\\& & 3& \color{blue}{-11} & \\ \hline &-3&\color{blue}{11}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -11 \right) } = \color{orangered}{ -8 } $

$$ \begin{array}{c|rrrr}-1&-3&8&\color{orangered}{ 3 }&-7\\& & 3& \color{orangered}{-11} & \\ \hline &-3&11&\color{orangered}{-8}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&-3&8&3&-7\\& & 3& -11& \color{blue}{8} \\ \hline &-3&11&\color{blue}{-8}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 8 } = \color{orangered}{ 1 } $

$$ \begin{array}{c|rrrr}-1&-3&8&3&\color{orangered}{ -7 }\\& & 3& -11& \color{orangered}{8} \\ \hline &\color{blue}{-3}&\color{blue}{11}&\color{blue}{-8}&\color{orangered}{1} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 1 }\right)$.

Synthetic Division Calculator