Check if $ x-1 $ is a factor of $ 3x^{4}-11x^{3}-3x^{2}-6x+8 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}1&3&-11&-3&-6&8\\& & 3& -8& -11& \color{black}{-17} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{-11}&\color{blue}{-17}&\color{orangered}{-9} \end{array} $$

Because the remainder $ \left( \color{red}{ -9 } \right) $ is not zero, we conclude that the $ x-1 $ is not a factor of $ 3x^{4}-11x^{3}-3x^{2}-6x+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 -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.

$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&-3&-6&8\\& & & & & \\ \hline &&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 3 }&-11&-3&-6&8\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&-3&-6&8\\& & \color{blue}{3} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}1&3&\color{orangered}{ -11 }&-3&-6&8\\& & \color{orangered}{3} & & & \\ \hline &3&\color{orangered}{-8}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -8 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&-3&-6&8\\& & 3& \color{blue}{-8} & & \\ \hline &3&\color{blue}{-8}&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}1&3&-11&\color{orangered}{ -3 }&-6&8\\& & 3& \color{orangered}{-8} & & \\ \hline &3&-8&\color{orangered}{-11}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ -11 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&-3&-6&8\\& & 3& -8& \color{blue}{-11} & \\ \hline &3&-8&\color{blue}{-11}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -11 \right) } = \color{orangered}{ -17 } $

$$ \begin{array}{c|rrrrr}1&3&-11&-3&\color{orangered}{ -6 }&8\\& & 3& -8& \color{orangered}{-11} & \\ \hline &3&-8&-11&\color{orangered}{-17}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -17 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{1}&3&-11&-3&-6&8\\& & 3& -8& -11& \color{blue}{-17} \\ \hline &3&-8&-11&\color{blue}{-17}& \end{array} $$

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

$$ \begin{array}{c|rrrrr}1&3&-11&-3&-6&\color{orangered}{ 8 }\\& & 3& -8& -11& \color{orangered}{-17} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{-11}&\color{blue}{-17}&\color{orangered}{-9} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -9 }\right)$.

Synthetic Division Calculator