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