Divide using synthetic division $$ ( 2x^{4}-13x^{3}+2x^{2}-21x+6 ) \div ( 3x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&2&-13&2&-21&6\\& & 2& -11& -9& \color{black}{-30} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{blue}{-9}&\color{blue}{-30}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-13x^{3}+2x^{2}-21x+6 }{ x-1 } = \color{blue}{2x^{3}-11x^{2}-9x-30} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-13&2&-21&6\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 2 }&-13&2&-21&6\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-13&2&-21&6\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 2 } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrr}1&2&\color{orangered}{ -13 }&2&-21&6\\& & \color{orangered}{2} & & & \\ \hline &2&\color{orangered}{-11}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ -11 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-13&2&-21&6\\& & 2& \color{blue}{-11} & & \\ \hline &2&\color{blue}{-11}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -11 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}1&2&-13&\color{orangered}{ 2 }&-21&6\\& & 2& \color{orangered}{-11} & & \\ \hline &2&-11&\color{orangered}{-9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-13&2&-21&6\\& & 2& -11& \color{blue}{-9} & \\ \hline &2&-11&\color{blue}{-9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrrr}1&2&-13&2&\color{orangered}{ -21 }&6\\& & 2& -11& \color{orangered}{-9} & \\ \hline &2&-11&-9&\color{orangered}{-30}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-13&2&-21&6\\& & 2& -11& -9& \color{blue}{-30} \\ \hline &2&-11&-9&\color{blue}{-30}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrr}1&2&-13&2&-21&\color{orangered}{ 6 }\\& & 2& -11& -9& \color{orangered}{-30} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{blue}{-9}&\color{blue}{-30}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-11x^{2}-9x-30 } $ with a remainder of $ \color{red}{ -24 } $.