Divide using synthetic division $$ ( 6x^{3}-x^{2}+4x-9 ) \div ( 2x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&6&-1&4&-9\\& & 18& 51& \color{black}{165} \\ \hline &\color{blue}{6}&\color{blue}{17}&\color{blue}{55}&\color{orangered}{156} \end{array} $$The solution is:
$$ \frac{ 6x^{3}-x^{2}+4x-9 }{ x-3 } = \color{blue}{6x^{2}+17x+55} ~+~ \frac{ \color{red}{ 156 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&-1&4&-9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 6 }&-1&4&-9\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&-1&4&-9\\& & \color{blue}{18} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 18 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrr}3&6&\color{orangered}{ -1 }&4&-9\\& & \color{orangered}{18} & & \\ \hline &6&\color{orangered}{17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 17 } = \color{blue}{ 51 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&-1&4&-9\\& & 18& \color{blue}{51} & \\ \hline &6&\color{blue}{17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 51 } = \color{orangered}{ 55 } $
$$ \begin{array}{c|rrrr}3&6&-1&\color{orangered}{ 4 }&-9\\& & 18& \color{orangered}{51} & \\ \hline &6&17&\color{orangered}{55}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 55 } = \color{blue}{ 165 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&6&-1&4&-9\\& & 18& 51& \color{blue}{165} \\ \hline &6&17&\color{blue}{55}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 165 } = \color{orangered}{ 156 } $
$$ \begin{array}{c|rrrr}3&6&-1&4&\color{orangered}{ -9 }\\& & 18& 51& \color{orangered}{165} \\ \hline &\color{blue}{6}&\color{blue}{17}&\color{blue}{55}&\color{orangered}{156} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+17x+55 } $ with a remainder of $ \color{red}{ 156 } $.