Divide using synthetic division $$ ( 3x^{3}-20x^{2}+21x-6 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&3&-20&21&-6\\& & 6& -28& \color{black}{-14} \\ \hline &\color{blue}{3}&\color{blue}{-14}&\color{blue}{-7}&\color{orangered}{-20} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-20x^{2}+21x-6 }{ x-2 } = \color{blue}{3x^{2}-14x-7} \color{red}{~-~} \frac{ \color{red}{ 20 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-20&21&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 3 }&-20&21&-6\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-20&21&-6\\& & \color{blue}{6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 6 } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrr}2&3&\color{orangered}{ -20 }&21&-6\\& & \color{orangered}{6} & & \\ \hline &3&\color{orangered}{-14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-20&21&-6\\& & 6& \color{blue}{-28} & \\ \hline &3&\color{blue}{-14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}2&3&-20&\color{orangered}{ 21 }&-6\\& & 6& \color{orangered}{-28} & \\ \hline &3&-14&\color{orangered}{-7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-20&21&-6\\& & 6& -28& \color{blue}{-14} \\ \hline &3&-14&\color{blue}{-7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrr}2&3&-20&21&\color{orangered}{ -6 }\\& & 6& -28& \color{orangered}{-14} \\ \hline &\color{blue}{3}&\color{blue}{-14}&\color{blue}{-7}&\color{orangered}{-20} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-14x-7 } $ with a remainder of $ \color{red}{ -20 } $.