Divide using synthetic division $$ ( 3x^{4}-9x^{3}-2x^{2}-3x+19 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&3&-9&-2&-3&19\\& & 9& 0& -6& \color{black}{-27} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{-2}&\color{blue}{-9}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-9x^{3}-2x^{2}-3x+19 }{ x-3 } = \color{blue}{3x^{3}-2x-9} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&-9&-2&-3&19\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 3 }&-9&-2&-3&19\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&-9&-2&-3&19\\& & \color{blue}{9} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 9 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}3&3&\color{orangered}{ -9 }&-2&-3&19\\& & \color{orangered}{9} & & & \\ \hline &3&\color{orangered}{0}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&-9&-2&-3&19\\& & 9& \color{blue}{0} & & \\ \hline &3&\color{blue}{0}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 0 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}3&3&-9&\color{orangered}{ -2 }&-3&19\\& & 9& \color{orangered}{0} & & \\ \hline &3&0&\color{orangered}{-2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&-9&-2&-3&19\\& & 9& 0& \color{blue}{-6} & \\ \hline &3&0&\color{blue}{-2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}3&3&-9&-2&\color{orangered}{ -3 }&19\\& & 9& 0& \color{orangered}{-6} & \\ \hline &3&0&-2&\color{orangered}{-9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&-9&-2&-3&19\\& & 9& 0& -6& \color{blue}{-27} \\ \hline &3&0&-2&\color{blue}{-9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}3&3&-9&-2&-3&\color{orangered}{ 19 }\\& & 9& 0& -6& \color{orangered}{-27} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{-2}&\color{blue}{-9}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-2x-9 } $ with a remainder of $ \color{red}{ -8 } $.