Check if $ 3x-4 $ is a factor of $ 9x^{4}-6x^{3}-2x^{2}+16x+8 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&9&-6&-2&16&8\\& & 36& 120& 472& \color{black}{1952} \\ \hline &\color{blue}{9}&\color{blue}{30}&\color{blue}{118}&\color{blue}{488}&\color{orangered}{1960} \end{array} $$Because the remainder $ \left( \color{red}{ 1960 } \right) $ is not zero, we conclude that the $ x-4 $ is not a factor of $ 9x^{4}-6x^{3}-2x^{2}+16x+8$.
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 -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-6&-2&16&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 9 }&-6&-2&16&8\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 9 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-6&-2&16&8\\& & \color{blue}{36} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 36 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrrr}4&9&\color{orangered}{ -6 }&-2&16&8\\& & \color{orangered}{36} & & & \\ \hline &9&\color{orangered}{30}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 30 } = \color{blue}{ 120 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-6&-2&16&8\\& & 36& \color{blue}{120} & & \\ \hline &9&\color{blue}{30}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 120 } = \color{orangered}{ 118 } $
$$ \begin{array}{c|rrrrr}4&9&-6&\color{orangered}{ -2 }&16&8\\& & 36& \color{orangered}{120} & & \\ \hline &9&30&\color{orangered}{118}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 118 } = \color{blue}{ 472 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-6&-2&16&8\\& & 36& 120& \color{blue}{472} & \\ \hline &9&30&\color{blue}{118}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 472 } = \color{orangered}{ 488 } $
$$ \begin{array}{c|rrrrr}4&9&-6&-2&\color{orangered}{ 16 }&8\\& & 36& 120& \color{orangered}{472} & \\ \hline &9&30&118&\color{orangered}{488}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 488 } = \color{blue}{ 1952 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-6&-2&16&8\\& & 36& 120& 472& \color{blue}{1952} \\ \hline &9&30&118&\color{blue}{488}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 1952 } = \color{orangered}{ 1960 } $
$$ \begin{array}{c|rrrrr}4&9&-6&-2&16&\color{orangered}{ 8 }\\& & 36& 120& 472& \color{orangered}{1952} \\ \hline &\color{blue}{9}&\color{blue}{30}&\color{blue}{118}&\color{blue}{488}&\color{orangered}{1960} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 1960 }\right)$.