Divide using synthetic division $$ ( 9x^{3}-6x^{2}+4x-4 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&9&-6&4&-4\\& & 27& 63& \color{black}{201} \\ \hline &\color{blue}{9}&\color{blue}{21}&\color{blue}{67}&\color{orangered}{197} \end{array} $$The solution is:
$$ \frac{ 9x^{3}-6x^{2}+4x-4 }{ x-3 } = \color{blue}{9x^{2}+21x+67} ~+~ \frac{ \color{red}{ 197 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-6&4&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 9 }&-6&4&-4\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 9 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-6&4&-4\\& & \color{blue}{27} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 27 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrr}3&9&\color{orangered}{ -6 }&4&-4\\& & \color{orangered}{27} & & \\ \hline &9&\color{orangered}{21}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 21 } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-6&4&-4\\& & 27& \color{blue}{63} & \\ \hline &9&\color{blue}{21}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 63 } = \color{orangered}{ 67 } $
$$ \begin{array}{c|rrrr}3&9&-6&\color{orangered}{ 4 }&-4\\& & 27& \color{orangered}{63} & \\ \hline &9&21&\color{orangered}{67}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 67 } = \color{blue}{ 201 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-6&4&-4\\& & 27& 63& \color{blue}{201} \\ \hline &9&21&\color{blue}{67}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 201 } = \color{orangered}{ 197 } $
$$ \begin{array}{c|rrrr}3&9&-6&4&\color{orangered}{ -4 }\\& & 27& 63& \color{orangered}{201} \\ \hline &\color{blue}{9}&\color{blue}{21}&\color{blue}{67}&\color{orangered}{197} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{2}+21x+67 } $ with a remainder of $ \color{red}{ 197 } $.