Divide using synthetic division $$ ( 3x^{3}-12x^{2}+16x-12 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&3&-12&16&-12\\& & 9& -9& \color{black}{21} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{7}&\color{orangered}{9} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-12x^{2}+16x-12 }{ x-3 } = \color{blue}{3x^{2}-3x+7} ~+~ \frac{ \color{red}{ 9 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-12&16&-12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 3 }&-12&16&-12\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-12&16&-12\\& & \color{blue}{9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 9 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}3&3&\color{orangered}{ -12 }&16&-12\\& & \color{orangered}{9} & & \\ \hline &3&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-12&16&-12\\& & 9& \color{blue}{-9} & \\ \hline &3&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}3&3&-12&\color{orangered}{ 16 }&-12\\& & 9& \color{orangered}{-9} & \\ \hline &3&-3&\color{orangered}{7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 7 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&-12&16&-12\\& & 9& -9& \color{blue}{21} \\ \hline &3&-3&\color{blue}{7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 21 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}3&3&-12&16&\color{orangered}{ -12 }\\& & 9& -9& \color{orangered}{21} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{7}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-3x+7 } $ with a remainder of $ \color{red}{ 9 } $.