Divide using synthetic division $$ ( 3x^{5}-3x^{4}+2x^{3}-x+9 ) \div ( x-20 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}20&3&-3&2&0&-1&9\\& & 60& 1140& 22840& 456800& \color{black}{9135980} \\ \hline &\color{blue}{3}&\color{blue}{57}&\color{blue}{1142}&\color{blue}{22840}&\color{blue}{456799}&\color{orangered}{9135989} \end{array} $$The solution is:
$$ \frac{ 3x^{5}-3x^{4}+2x^{3}-x+9 }{ x-20 } = \color{blue}{3x^{4}+57x^{3}+1142x^{2}+22840x+456799} ~+~ \frac{ \color{red}{ 9135989 } }{ x-20 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -20 = 0 $ ( $ x = \color{blue}{ 20 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{20}&3&-3&2&0&-1&9\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}20&\color{orangered}{ 3 }&-3&2&0&-1&9\\& & & & & & \\ \hline &\color{orangered}{3}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 3 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{20}&3&-3&2&0&-1&9\\& & \color{blue}{60} & & & & \\ \hline &\color{blue}{3}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 60 } = \color{orangered}{ 57 } $
$$ \begin{array}{c|rrrrrr}20&3&\color{orangered}{ -3 }&2&0&-1&9\\& & \color{orangered}{60} & & & & \\ \hline &3&\color{orangered}{57}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 57 } = \color{blue}{ 1140 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{20}&3&-3&2&0&-1&9\\& & 60& \color{blue}{1140} & & & \\ \hline &3&\color{blue}{57}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 1140 } = \color{orangered}{ 1142 } $
$$ \begin{array}{c|rrrrrr}20&3&-3&\color{orangered}{ 2 }&0&-1&9\\& & 60& \color{orangered}{1140} & & & \\ \hline &3&57&\color{orangered}{1142}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 1142 } = \color{blue}{ 22840 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{20}&3&-3&2&0&-1&9\\& & 60& 1140& \color{blue}{22840} & & \\ \hline &3&57&\color{blue}{1142}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 22840 } = \color{orangered}{ 22840 } $
$$ \begin{array}{c|rrrrrr}20&3&-3&2&\color{orangered}{ 0 }&-1&9\\& & 60& 1140& \color{orangered}{22840} & & \\ \hline &3&57&1142&\color{orangered}{22840}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 22840 } = \color{blue}{ 456800 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{20}&3&-3&2&0&-1&9\\& & 60& 1140& 22840& \color{blue}{456800} & \\ \hline &3&57&1142&\color{blue}{22840}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 456800 } = \color{orangered}{ 456799 } $
$$ \begin{array}{c|rrrrrr}20&3&-3&2&0&\color{orangered}{ -1 }&9\\& & 60& 1140& 22840& \color{orangered}{456800} & \\ \hline &3&57&1142&22840&\color{orangered}{456799}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 456799 } = \color{blue}{ 9135980 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{20}&3&-3&2&0&-1&9\\& & 60& 1140& 22840& 456800& \color{blue}{9135980} \\ \hline &3&57&1142&22840&\color{blue}{456799}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 9135980 } = \color{orangered}{ 9135989 } $
$$ \begin{array}{c|rrrrrr}20&3&-3&2&0&-1&\color{orangered}{ 9 }\\& & 60& 1140& 22840& 456800& \color{orangered}{9135980} \\ \hline &\color{blue}{3}&\color{blue}{57}&\color{blue}{1142}&\color{blue}{22840}&\color{blue}{456799}&\color{orangered}{9135989} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{4}+57x^{3}+1142x^{2}+22840x+456799 } $ with a remainder of $ \color{red}{ 9135989 } $.