Divide using synthetic division $$ ( -3x^{3}+12x^{2}-11x+6 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&-3&12&-11&6\\& & -9& 9& \color{black}{-6} \\ \hline &\color{blue}{-3}&\color{blue}{3}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -3x^{3}+12x^{2}-11x+6 }{ x-3 } = \color{blue}{-3x^{2}+3x-2} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&12&-11&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ -3 }&12&-11&6\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&12&-11&6\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}3&-3&\color{orangered}{ 12 }&-11&6\\& & \color{orangered}{-9} & & \\ \hline &-3&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&12&-11&6\\& & -9& \color{blue}{9} & \\ \hline &-3&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 9 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}3&-3&12&\color{orangered}{ -11 }&6\\& & -9& \color{orangered}{9} & \\ \hline &-3&3&\color{orangered}{-2}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&12&-11&6\\& & -9& 9& \color{blue}{-6} \\ \hline &-3&3&\color{blue}{-2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}3&-3&12&-11&\color{orangered}{ 6 }\\& & -9& 9& \color{orangered}{-6} \\ \hline &\color{blue}{-3}&\color{blue}{3}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{2}+3x-2 } $ with a remainder of $ \color{red}{ 0 } $.