Divide using synthetic division $$ ( 32768x^{5}+16807 ) \div ( 24x+21 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-21&32768&0&0&0&0&16807\\& & -688128& 14450688& -303464448& 6372753408& \color{black}{-133827821568} \\ \hline &\color{blue}{32768}&\color{blue}{-688128}&\color{blue}{14450688}&\color{blue}{-303464448}&\color{blue}{6372753408}&\color{orangered}{-133827804761} \end{array} $$The solution is:
$$ \frac{ 32768x^{5}+16807 }{ x+21 } = \color{blue}{32768x^{4}-688128x^{3}+14450688x^{2}-303464448x+6372753408} \color{red}{~-~} \frac{ \color{red}{ 133827804761 } }{ x+21 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 21 = 0 $ ( $ x = \color{blue}{ -21 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-21}&32768&0&0&0&0&16807\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-21&\color{orangered}{ 32768 }&0&0&0&0&16807\\& & & & & & \\ \hline &\color{orangered}{32768}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -21 } \cdot \color{blue}{ 32768 } = \color{blue}{ -688128 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-21}&32768&0&0&0&0&16807\\& & \color{blue}{-688128} & & & & \\ \hline &\color{blue}{32768}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -688128 \right) } = \color{orangered}{ -688128 } $
$$ \begin{array}{c|rrrrrr}-21&32768&\color{orangered}{ 0 }&0&0&0&16807\\& & \color{orangered}{-688128} & & & & \\ \hline &32768&\color{orangered}{-688128}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -21 } \cdot \color{blue}{ \left( -688128 \right) } = \color{blue}{ 14450688 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-21}&32768&0&0&0&0&16807\\& & -688128& \color{blue}{14450688} & & & \\ \hline &32768&\color{blue}{-688128}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 14450688 } = \color{orangered}{ 14450688 } $
$$ \begin{array}{c|rrrrrr}-21&32768&0&\color{orangered}{ 0 }&0&0&16807\\& & -688128& \color{orangered}{14450688} & & & \\ \hline &32768&-688128&\color{orangered}{14450688}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -21 } \cdot \color{blue}{ 14450688 } = \color{blue}{ -303464448 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-21}&32768&0&0&0&0&16807\\& & -688128& 14450688& \color{blue}{-303464448} & & \\ \hline &32768&-688128&\color{blue}{14450688}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -303464448 \right) } = \color{orangered}{ -303464448 } $
$$ \begin{array}{c|rrrrrr}-21&32768&0&0&\color{orangered}{ 0 }&0&16807\\& & -688128& 14450688& \color{orangered}{-303464448} & & \\ \hline &32768&-688128&14450688&\color{orangered}{-303464448}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -21 } \cdot \color{blue}{ \left( -303464448 \right) } = \color{blue}{ 6372753408 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-21}&32768&0&0&0&0&16807\\& & -688128& 14450688& -303464448& \color{blue}{6372753408} & \\ \hline &32768&-688128&14450688&\color{blue}{-303464448}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6372753408 } = \color{orangered}{ 6372753408 } $
$$ \begin{array}{c|rrrrrr}-21&32768&0&0&0&\color{orangered}{ 0 }&16807\\& & -688128& 14450688& -303464448& \color{orangered}{6372753408} & \\ \hline &32768&-688128&14450688&-303464448&\color{orangered}{6372753408}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -21 } \cdot \color{blue}{ 6372753408 } = \color{blue}{ -133827821568 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-21}&32768&0&0&0&0&16807\\& & -688128& 14450688& -303464448& 6372753408& \color{blue}{-133827821568} \\ \hline &32768&-688128&14450688&-303464448&\color{blue}{6372753408}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 16807 } + \color{orangered}{ \left( -133827821568 \right) } = \color{orangered}{ -133827804761 } $
$$ \begin{array}{c|rrrrrr}-21&32768&0&0&0&0&\color{orangered}{ 16807 }\\& & -688128& 14450688& -303464448& 6372753408& \color{orangered}{-133827821568} \\ \hline &\color{blue}{32768}&\color{blue}{-688128}&\color{blue}{14450688}&\color{blue}{-303464448}&\color{blue}{6372753408}&\color{orangered}{-133827804761} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 32768x^{4}-688128x^{3}+14450688x^{2}-303464448x+6372753408 } $ with a remainder of $ \color{red}{ -133827804761 } $.