Check if $ x-3 $ is a factor of $ -2x^{2}+27x-94 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-2&27&-94\\& & -6& \color{black}{63} \\ \hline &\color{blue}{-2}&\color{blue}{21}&\color{orangered}{-31} \end{array} $$Because the remainder $ \left( \color{red}{ -31 } \right) $ is not zero, we conclude that the $ x-3 $ is not a factor of $ -2x^{2}+27x-94$.
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.
$$ \begin{array}{c|rrr}\color{blue}{3}&-2&27&-94\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -2 }&27&-94\\& & & \\ \hline &\color{orangered}{-2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-2&27&-94\\& & \color{blue}{-6} & \\ \hline &\color{blue}{-2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrr}3&-2&\color{orangered}{ 27 }&-94\\& & \color{orangered}{-6} & \\ \hline &-2&\color{orangered}{21}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 21 } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-2&27&-94\\& & -6& \color{blue}{63} \\ \hline &-2&\color{blue}{21}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -94 } + \color{orangered}{ 63 } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrr}3&-2&27&\color{orangered}{ -94 }\\& & -6& \color{orangered}{63} \\ \hline &\color{blue}{-2}&\color{blue}{21}&\color{orangered}{-31} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -31 }\right)$.