Check if $ x-3 $ is a factor of $ 3x^{3}-3x^{2}-108 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}3&3&-3&0&-108\\& & 9& 18& \color{black}{54} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{blue}{18}&\color{orangered}{-54} \end{array} $$

Because the remainder $ \left( \color{red}{ -54 } \right) $ is not zero, we conclude that the $ x-3 $ is not a factor of $ 3x^{3}-3x^{2}-108$.

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 -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-3&0&-108\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}3&\color{orangered}{ 3 }&-3&0&-108\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.

$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-3&0&-108\\& & \color{blue}{9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 9 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrrr}3&3&\color{orangered}{ -3 }&0&-108\\& & \color{orangered}{9} & & \\ \hline &3&\color{orangered}{6}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.

$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-3&0&-108\\& & 9& \color{blue}{18} & \\ \hline &3&\color{blue}{6}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $

$$ \begin{array}{c|rrrr}3&3&-3&\color{orangered}{ 0 }&-108\\& & 9& \color{orangered}{18} & \\ \hline &3&6&\color{orangered}{18}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 18 } = \color{blue}{ 54 } $.

$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-3&0&-108\\& & 9& 18& \color{blue}{54} \\ \hline &3&6&\color{blue}{18}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -108 } + \color{orangered}{ 54 } = \color{orangered}{ -54 } $

$$ \begin{array}{c|rrrr}3&3&-3&0&\color{orangered}{ -108 }\\& & 9& 18& \color{orangered}{54} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{blue}{18}&\color{orangered}{-54} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -54 }\right)$.

Synthetic Division Calculator