Check if $ x-3 $ is a factor of $ 2x^{3}-31x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&2&0&-31&2\\& & 6& 18& \color{black}{-39} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{-13}&\color{orangered}{-37} \end{array} $$Because the remainder $ \left( \color{red}{ -37 } \right) $ is not zero, we conclude that the $ x-3 $ is not a factor of $ 2x^{3}-31x+2$.
Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&-31&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 2 }&0&-31&2\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&-31&2\\& & \color{blue}{6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}3&2&\color{orangered}{ 0 }&-31&2\\& & \color{orangered}{6} & & \\ \hline &2&\color{orangered}{6}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&-31&2\\& & 6& \color{blue}{18} & \\ \hline &2&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -31 } + \color{orangered}{ 18 } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}3&2&0&\color{orangered}{ -31 }&2\\& & 6& \color{orangered}{18} & \\ \hline &2&6&\color{orangered}{-13}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ -39 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&-31&2\\& & 6& 18& \color{blue}{-39} \\ \hline &2&6&\color{blue}{-13}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -39 \right) } = \color{orangered}{ -37 } $
$$ \begin{array}{c|rrrr}3&2&0&-31&\color{orangered}{ 2 }\\& & 6& 18& \color{orangered}{-39} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{-13}&\color{orangered}{-37} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -37 }\right)$.