Divide using synthetic division $$ ( 3x^{3}-20x^{2}+12x ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&3&-20&12&0\\& & 18& -12& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{blue}{0}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-20x^{2}+12x }{ x-6 } = \color{blue}{3x^{2}-2x} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&-20&12&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 3 }&-20&12&0\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 3 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&-20&12&0\\& & \color{blue}{18} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 18 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}6&3&\color{orangered}{ -20 }&12&0\\& & \color{orangered}{18} & & \\ \hline &3&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&-20&12&0\\& & 18& \color{blue}{-12} & \\ \hline &3&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}6&3&-20&\color{orangered}{ 12 }&0\\& & 18& \color{orangered}{-12} & \\ \hline &3&-2&\color{orangered}{0}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&-20&12&0\\& & 18& -12& \color{blue}{0} \\ \hline &3&-2&\color{blue}{0}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}6&3&-20&12&\color{orangered}{ 0 }\\& & 18& -12& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{blue}{0}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-2x } $ with a remainder of $ \color{red}{ 0 } $.