Divide using synthetic division $$ ( x^{5}+3x^{4}+2x^{3}+3x^{2} ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}3&1&3&2&3&0&0\\& & 3& 18& 60& 189& \color{black}{567} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{20}&\color{blue}{63}&\color{blue}{189}&\color{orangered}{567} \end{array} $$The solution is:
$$ \frac{ x^{5}+3x^{4}+2x^{3}+3x^{2} }{ x-3 } = \color{blue}{x^{4}+6x^{3}+20x^{2}+63x+189} ~+~ \frac{ \color{red}{ 567 } }{ x-3 } $$Explanation
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&3&2&3&0&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}3&\color{orangered}{ 1 }&3&2&3&0&0\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&3&2&3&0&0\\& & \color{blue}{3} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 3 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrr}3&1&\color{orangered}{ 3 }&2&3&0&0\\& & \color{orangered}{3} & & & & \\ \hline &1&\color{orangered}{6}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&3&2&3&0&0\\& & 3& \color{blue}{18} & & & \\ \hline &1&\color{blue}{6}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 18 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrrrr}3&1&3&\color{orangered}{ 2 }&3&0&0\\& & 3& \color{orangered}{18} & & & \\ \hline &1&6&\color{orangered}{20}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 20 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&3&2&3&0&0\\& & 3& 18& \color{blue}{60} & & \\ \hline &1&6&\color{blue}{20}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 60 } = \color{orangered}{ 63 } $
$$ \begin{array}{c|rrrrrr}3&1&3&2&\color{orangered}{ 3 }&0&0\\& & 3& 18& \color{orangered}{60} & & \\ \hline &1&6&20&\color{orangered}{63}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 63 } = \color{blue}{ 189 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&3&2&3&0&0\\& & 3& 18& 60& \color{blue}{189} & \\ \hline &1&6&20&\color{blue}{63}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 189 } = \color{orangered}{ 189 } $
$$ \begin{array}{c|rrrrrr}3&1&3&2&3&\color{orangered}{ 0 }&0\\& & 3& 18& 60& \color{orangered}{189} & \\ \hline &1&6&20&63&\color{orangered}{189}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 189 } = \color{blue}{ 567 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&3&2&3&0&0\\& & 3& 18& 60& 189& \color{blue}{567} \\ \hline &1&6&20&63&\color{blue}{189}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 567 } = \color{orangered}{ 567 } $
$$ \begin{array}{c|rrrrrr}3&1&3&2&3&0&\color{orangered}{ 0 }\\& & 3& 18& 60& 189& \color{orangered}{567} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{20}&\color{blue}{63}&\color{blue}{189}&\color{orangered}{567} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+6x^{3}+20x^{2}+63x+189 } $ with a remainder of $ \color{red}{ 567 } $.