Check if $ 2x-3 $ is a factor of $ 32x^{5}+32x^{4}-16x^{3}+4x^{2}-16x-336 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}3&32&32&-16&4&-16&-336\\& & 96& 384& 1104& 3324& \color{black}{9924} \\ \hline &\color{blue}{32}&\color{blue}{128}&\color{blue}{368}&\color{blue}{1108}&\color{blue}{3308}&\color{orangered}{9588} \end{array} $$Because the remainder $ \left( \color{red}{ 9588 } \right) $ is not zero, we conclude that the $ x-3 $ is not a factor of $ 32x^{5}+32x^{4}-16x^{3}+4x^{2}-16x-336$.
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&32&32&-16&4&-16&-336\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}3&\color{orangered}{ 32 }&32&-16&4&-16&-336\\& & & & & & \\ \hline &\color{orangered}{32}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 32 } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&32&32&-16&4&-16&-336\\& & \color{blue}{96} & & & & \\ \hline &\color{blue}{32}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ 96 } = \color{orangered}{ 128 } $
$$ \begin{array}{c|rrrrrr}3&32&\color{orangered}{ 32 }&-16&4&-16&-336\\& & \color{orangered}{96} & & & & \\ \hline &32&\color{orangered}{128}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 128 } = \color{blue}{ 384 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&32&32&-16&4&-16&-336\\& & 96& \color{blue}{384} & & & \\ \hline &32&\color{blue}{128}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 384 } = \color{orangered}{ 368 } $
$$ \begin{array}{c|rrrrrr}3&32&32&\color{orangered}{ -16 }&4&-16&-336\\& & 96& \color{orangered}{384} & & & \\ \hline &32&128&\color{orangered}{368}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 368 } = \color{blue}{ 1104 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&32&32&-16&4&-16&-336\\& & 96& 384& \color{blue}{1104} & & \\ \hline &32&128&\color{blue}{368}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 1104 } = \color{orangered}{ 1108 } $
$$ \begin{array}{c|rrrrrr}3&32&32&-16&\color{orangered}{ 4 }&-16&-336\\& & 96& 384& \color{orangered}{1104} & & \\ \hline &32&128&368&\color{orangered}{1108}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1108 } = \color{blue}{ 3324 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&32&32&-16&4&-16&-336\\& & 96& 384& 1104& \color{blue}{3324} & \\ \hline &32&128&368&\color{blue}{1108}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 3324 } = \color{orangered}{ 3308 } $
$$ \begin{array}{c|rrrrrr}3&32&32&-16&4&\color{orangered}{ -16 }&-336\\& & 96& 384& 1104& \color{orangered}{3324} & \\ \hline &32&128&368&1108&\color{orangered}{3308}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3308 } = \color{blue}{ 9924 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&32&32&-16&4&-16&-336\\& & 96& 384& 1104& 3324& \color{blue}{9924} \\ \hline &32&128&368&1108&\color{blue}{3308}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -336 } + \color{orangered}{ 9924 } = \color{orangered}{ 9588 } $
$$ \begin{array}{c|rrrrrr}3&32&32&-16&4&-16&\color{orangered}{ -336 }\\& & 96& 384& 1104& 3324& \color{orangered}{9924} \\ \hline &\color{blue}{32}&\color{blue}{128}&\color{blue}{368}&\color{blue}{1108}&\color{blue}{3308}&\color{orangered}{9588} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 9588 }\right)$.