Divide using synthetic division $$ ( 32768x^{5}+16807 ) \div ( 3x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-7&32768&0&0&0&0&16807\\& & -229376& 1605632& -11239424& 78675968& \color{black}{-550731776} \\ \hline &\color{blue}{32768}&\color{blue}{-229376}&\color{blue}{1605632}&\color{blue}{-11239424}&\color{blue}{78675968}&\color{orangered}{-550714969} \end{array} $$The solution is:
$$ \frac{ 32768x^{5}+16807 }{ x+7 } = \color{blue}{32768x^{4}-229376x^{3}+1605632x^{2}-11239424x+78675968} \color{red}{~-~} \frac{ \color{red}{ 550714969 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-7}&32768&0&0&0&0&16807\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-7&\color{orangered}{ 32768 }&0&0&0&0&16807\\& & & & & & \\ \hline &\color{orangered}{32768}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 32768 } = \color{blue}{ -229376 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-7}&32768&0&0&0&0&16807\\& & \color{blue}{-229376} & & & & \\ \hline &\color{blue}{32768}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -229376 \right) } = \color{orangered}{ -229376 } $
$$ \begin{array}{c|rrrrrr}-7&32768&\color{orangered}{ 0 }&0&0&0&16807\\& & \color{orangered}{-229376} & & & & \\ \hline &32768&\color{orangered}{-229376}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -229376 \right) } = \color{blue}{ 1605632 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-7}&32768&0&0&0&0&16807\\& & -229376& \color{blue}{1605632} & & & \\ \hline &32768&\color{blue}{-229376}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1605632 } = \color{orangered}{ 1605632 } $
$$ \begin{array}{c|rrrrrr}-7&32768&0&\color{orangered}{ 0 }&0&0&16807\\& & -229376& \color{orangered}{1605632} & & & \\ \hline &32768&-229376&\color{orangered}{1605632}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 1605632 } = \color{blue}{ -11239424 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-7}&32768&0&0&0&0&16807\\& & -229376& 1605632& \color{blue}{-11239424} & & \\ \hline &32768&-229376&\color{blue}{1605632}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -11239424 \right) } = \color{orangered}{ -11239424 } $
$$ \begin{array}{c|rrrrrr}-7&32768&0&0&\color{orangered}{ 0 }&0&16807\\& & -229376& 1605632& \color{orangered}{-11239424} & & \\ \hline &32768&-229376&1605632&\color{orangered}{-11239424}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -11239424 \right) } = \color{blue}{ 78675968 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-7}&32768&0&0&0&0&16807\\& & -229376& 1605632& -11239424& \color{blue}{78675968} & \\ \hline &32768&-229376&1605632&\color{blue}{-11239424}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 78675968 } = \color{orangered}{ 78675968 } $
$$ \begin{array}{c|rrrrrr}-7&32768&0&0&0&\color{orangered}{ 0 }&16807\\& & -229376& 1605632& -11239424& \color{orangered}{78675968} & \\ \hline &32768&-229376&1605632&-11239424&\color{orangered}{78675968}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 78675968 } = \color{blue}{ -550731776 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-7}&32768&0&0&0&0&16807\\& & -229376& 1605632& -11239424& 78675968& \color{blue}{-550731776} \\ \hline &32768&-229376&1605632&-11239424&\color{blue}{78675968}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 16807 } + \color{orangered}{ \left( -550731776 \right) } = \color{orangered}{ -550714969 } $
$$ \begin{array}{c|rrrrrr}-7&32768&0&0&0&0&\color{orangered}{ 16807 }\\& & -229376& 1605632& -11239424& 78675968& \color{orangered}{-550731776} \\ \hline &\color{blue}{32768}&\color{blue}{-229376}&\color{blue}{1605632}&\color{blue}{-11239424}&\color{blue}{78675968}&\color{orangered}{-550714969} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 32768x^{4}-229376x^{3}+1605632x^{2}-11239424x+78675968 } $ with a remainder of $ \color{red}{ -550714969 } $.