Check if $ x-8 $ is a factor of $ 9x^{4}-6x^{3}-2x^{2}+16x+8 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&9&-6&-2&16&8\\& & 72& 528& 4208& \color{black}{33792} \\ \hline &\color{blue}{9}&\color{blue}{66}&\color{blue}{526}&\color{blue}{4224}&\color{orangered}{33800} \end{array} $$Because the remainder $ \left( \color{red}{ 33800 } \right) $ is not zero, we conclude that the $ x-8 $ 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 -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&9&-6&-2&16&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 9 }&-6&-2&16&8\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 9 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&9&-6&-2&16&8\\& & \color{blue}{72} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 72 } = \color{orangered}{ 66 } $
$$ \begin{array}{c|rrrrr}8&9&\color{orangered}{ -6 }&-2&16&8\\& & \color{orangered}{72} & & & \\ \hline &9&\color{orangered}{66}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 66 } = \color{blue}{ 528 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&9&-6&-2&16&8\\& & 72& \color{blue}{528} & & \\ \hline &9&\color{blue}{66}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 528 } = \color{orangered}{ 526 } $
$$ \begin{array}{c|rrrrr}8&9&-6&\color{orangered}{ -2 }&16&8\\& & 72& \color{orangered}{528} & & \\ \hline &9&66&\color{orangered}{526}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 526 } = \color{blue}{ 4208 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&9&-6&-2&16&8\\& & 72& 528& \color{blue}{4208} & \\ \hline &9&66&\color{blue}{526}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 4208 } = \color{orangered}{ 4224 } $
$$ \begin{array}{c|rrrrr}8&9&-6&-2&\color{orangered}{ 16 }&8\\& & 72& 528& \color{orangered}{4208} & \\ \hline &9&66&526&\color{orangered}{4224}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 4224 } = \color{blue}{ 33792 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&9&-6&-2&16&8\\& & 72& 528& 4208& \color{blue}{33792} \\ \hline &9&66&526&\color{blue}{4224}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 33792 } = \color{orangered}{ 33800 } $
$$ \begin{array}{c|rrrrr}8&9&-6&-2&16&\color{orangered}{ 8 }\\& & 72& 528& 4208& \color{orangered}{33792} \\ \hline &\color{blue}{9}&\color{blue}{66}&\color{blue}{526}&\color{blue}{4224}&\color{orangered}{33800} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 33800 }\right)$.