Divide using synthetic division $$ ( 4x^{3}-20x^{2}+21x-6 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&4&-20&21&-6\\& & 8& -24& \color{black}{-6} \\ \hline &\color{blue}{4}&\color{blue}{-12}&\color{blue}{-3}&\color{orangered}{-12} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-20x^{2}+21x-6 }{ x-2 } = \color{blue}{4x^{2}-12x-3} \color{red}{~-~} \frac{ \color{red}{ 12 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-20&21&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 4 }&-20&21&-6\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-20&21&-6\\& & \color{blue}{8} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 8 } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}2&4&\color{orangered}{ -20 }&21&-6\\& & \color{orangered}{8} & & \\ \hline &4&\color{orangered}{-12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-20&21&-6\\& & 8& \color{blue}{-24} & \\ \hline &4&\color{blue}{-12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}2&4&-20&\color{orangered}{ 21 }&-6\\& & 8& \color{orangered}{-24} & \\ \hline &4&-12&\color{orangered}{-3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-20&21&-6\\& & 8& -24& \color{blue}{-6} \\ \hline &4&-12&\color{blue}{-3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}2&4&-20&21&\color{orangered}{ -6 }\\& & 8& -24& \color{orangered}{-6} \\ \hline &\color{blue}{4}&\color{blue}{-12}&\color{blue}{-3}&\color{orangered}{-12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-12x-3 } $ with a remainder of $ \color{red}{ -12 } $.