Divide using synthetic division $$ ( 3x^{3}+2x^{2}-9x+6 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&3&2&-9&6\\& & 9& 33& \color{black}{72} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{blue}{24}&\color{orangered}{78} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+2x^{2}-9x+6 }{ x-3 } = \color{blue}{3x^{2}+11x+24} ~+~ \frac{ \color{red}{ 78 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&2&-9&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 3 }&2&-9&6\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&2&-9&6\\& & \color{blue}{9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 9 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}3&3&\color{orangered}{ 2 }&-9&6\\& & \color{orangered}{9} & & \\ \hline &3&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 11 } = \color{blue}{ 33 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&2&-9&6\\& & 9& \color{blue}{33} & \\ \hline &3&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 33 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}3&3&2&\color{orangered}{ -9 }&6\\& & 9& \color{orangered}{33} & \\ \hline &3&11&\color{orangered}{24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 24 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&2&-9&6\\& & 9& 33& \color{blue}{72} \\ \hline &3&11&\color{blue}{24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 72 } = \color{orangered}{ 78 } $
$$ \begin{array}{c|rrrr}3&3&2&-9&\color{orangered}{ 6 }\\& & 9& 33& \color{orangered}{72} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{blue}{24}&\color{orangered}{78} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+11x+24 } $ with a remainder of $ \color{red}{ 78 } $.