Divide using synthetic division $$ ( x^{4}-3x^{3}+1.5x^{2}-6x+14 ) \div ( x-2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&1&-3&\dfrac{ 3 }{ 2 }&-6&14\\& & 2& -2& -1& \color{black}{-14} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-\dfrac{ 1 }{ 2 }}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-3x^{3}+\dfrac{ 3 }{ 2 }x^{2}-6x+14 }{ x-2 } = \color{blue}{x^{3}-x^{2}-\dfrac{ 1 }{ 2 }x-7} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-3&\frac{ 3 }{ 2 }&-6&14\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 1 }&-3&\frac{ 3 }{ 2 }&-6&14\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-3&\frac{ 3 }{ 2 }&-6&14\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 2 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}2&1&\color{orangered}{ -3 }&\frac{ 3 }{ 2 }&-6&14\\& & \color{orangered}{2} & & & \\ \hline &1&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-3&\frac{ 3 }{ 2 }&-6&14\\& & 2& \color{blue}{-2} & & \\ \hline &1&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ \frac{ 3 }{ 2 } } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -\frac{ 1 }{ 2 } } $
$$ \begin{array}{c|rrrrr}2&1&-3&\color{orangered}{ \frac{ 3 }{ 2 } }&-6&14\\& & 2& \color{orangered}{-2} & & \\ \hline &1&-1&\color{orangered}{-\frac{ 1 }{ 2 }}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -\frac{ 1 }{ 2 } \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-3&\frac{ 3 }{ 2 }&-6&14\\& & 2& -2& \color{blue}{-1} & \\ \hline &1&-1&\color{blue}{-\frac{ 1 }{ 2 }}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}2&1&-3&\frac{ 3 }{ 2 }&\color{orangered}{ -6 }&14\\& & 2& -2& \color{orangered}{-1} & \\ \hline &1&-1&-\frac{ 1 }{ 2 }&\color{orangered}{-7}& \end{array} $$Step 8 : 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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-3&\frac{ 3 }{ 2 }&-6&14\\& & 2& -2& -1& \color{blue}{-14} \\ \hline &1&-1&-\frac{ 1 }{ 2 }&\color{blue}{-7}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}2&1&-3&\frac{ 3 }{ 2 }&-6&\color{orangered}{ 14 }\\& & 2& -2& -1& \color{orangered}{-14} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-\frac{ 1 }{ 2 }}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-x^{2}-\frac{ 1 }{ 2 }x-7 } $ with a remainder of $ \color{red}{ 0 } $.