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