Divide using synthetic division $$ ( -0.01x^{5}+0.04x^{4}+0.35x^{3}-1.18x^{2}-3.2x+8 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & -\frac{ 1 }{ 50 }& \frac{ 1 }{ 25 }& \frac{ 39 }{ 50 }& -\frac{ 4 }{ 5 }& \color{black}{-8} \\ \hline &\color{blue}{-\frac{ 1 }{ 100 }}&\color{blue}{\frac{ 1 }{ 50 }}&\color{blue}{\frac{ 39 }{ 100 }}&\color{blue}{-\frac{ 2 }{ 5 }}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -\frac{ 1 }{ 100 }x^{5}+\frac{ 1 }{ 25 }x^{4}+\frac{ 7 }{ 20 }x^{3}-\frac{ 59 }{ 50 }x^{2}-\frac{ 16 }{ 5 }x+8 }{ x-2 } = \color{blue}{-\frac{ 1 }{ 100 }x^{4}+\frac{ 1 }{ 50 }x^{3}+\frac{ 39 }{ 100 }x^{2}-\frac{ 2 }{ 5 }x-4} $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ -\frac{ 1 }{ 100 } }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & & & & & \\ \hline &\color{orangered}{-\frac{ 1 }{ 100 }}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -\frac{ 1 }{ 100 } \right) } = \color{blue}{ -\frac{ 1 }{ 50 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & \color{blue}{-\frac{ 1 }{ 50 }} & & & & \\ \hline &\color{blue}{-\frac{ 1 }{ 100 }}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ \frac{ 1 }{ 25 } } + \color{orangered}{ \left( -\frac{ 1 }{ 50 } \right) } = \color{orangered}{ \frac{ 1 }{ 50 } } $
$$ \begin{array}{c|rrrrrr}2&-\frac{ 1 }{ 100 }&\color{orangered}{ \frac{ 1 }{ 25 } }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & \color{orangered}{-\frac{ 1 }{ 50 }} & & & & \\ \hline &-\frac{ 1 }{ 100 }&\color{orangered}{\frac{ 1 }{ 50 }}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \frac{ 1 }{ 50 } } = \color{blue}{ \frac{ 1 }{ 25 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & -\frac{ 1 }{ 50 }& \color{blue}{\frac{ 1 }{ 25 }} & & & \\ \hline &-\frac{ 1 }{ 100 }&\color{blue}{\frac{ 1 }{ 50 }}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ \frac{ 7 }{ 20 } } + \color{orangered}{ \frac{ 1 }{ 25 } } = \color{orangered}{ \frac{ 39 }{ 100 } } $
$$ \begin{array}{c|rrrrrr}2&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\color{orangered}{ \frac{ 7 }{ 20 } }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & -\frac{ 1 }{ 50 }& \color{orangered}{\frac{ 1 }{ 25 }} & & & \\ \hline &-\frac{ 1 }{ 100 }&\frac{ 1 }{ 50 }&\color{orangered}{\frac{ 39 }{ 100 }}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \frac{ 39 }{ 100 } } = \color{blue}{ \frac{ 39 }{ 50 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & -\frac{ 1 }{ 50 }& \frac{ 1 }{ 25 }& \color{blue}{\frac{ 39 }{ 50 }} & & \\ \hline &-\frac{ 1 }{ 100 }&\frac{ 1 }{ 50 }&\color{blue}{\frac{ 39 }{ 100 }}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -\frac{ 59 }{ 50 } } + \color{orangered}{ \frac{ 39 }{ 50 } } = \color{orangered}{ -\frac{ 2 }{ 5 } } $
$$ \begin{array}{c|rrrrrr}2&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&\color{orangered}{ -\frac{ 59 }{ 50 } }&-\frac{ 16 }{ 5 }&8\\& & -\frac{ 1 }{ 50 }& \frac{ 1 }{ 25 }& \color{orangered}{\frac{ 39 }{ 50 }} & & \\ \hline &-\frac{ 1 }{ 100 }&\frac{ 1 }{ 50 }&\frac{ 39 }{ 100 }&\color{orangered}{-\frac{ 2 }{ 5 }}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -\frac{ 2 }{ 5 } \right) } = \color{blue}{ -\frac{ 4 }{ 5 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & -\frac{ 1 }{ 50 }& \frac{ 1 }{ 25 }& \frac{ 39 }{ 50 }& \color{blue}{-\frac{ 4 }{ 5 }} & \\ \hline &-\frac{ 1 }{ 100 }&\frac{ 1 }{ 50 }&\frac{ 39 }{ 100 }&\color{blue}{-\frac{ 2 }{ 5 }}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -\frac{ 16 }{ 5 } } + \color{orangered}{ \left( -\frac{ 4 }{ 5 } \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}2&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&\color{orangered}{ -\frac{ 16 }{ 5 } }&8\\& & -\frac{ 1 }{ 50 }& \frac{ 1 }{ 25 }& \frac{ 39 }{ 50 }& \color{orangered}{-\frac{ 4 }{ 5 }} & \\ \hline &-\frac{ 1 }{ 100 }&\frac{ 1 }{ 50 }&\frac{ 39 }{ 100 }&-\frac{ 2 }{ 5 }&\color{orangered}{-4}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&8\\& & -\frac{ 1 }{ 50 }& \frac{ 1 }{ 25 }& \frac{ 39 }{ 50 }& -\frac{ 4 }{ 5 }& \color{blue}{-8} \\ \hline &-\frac{ 1 }{ 100 }&\frac{ 1 }{ 50 }&\frac{ 39 }{ 100 }&-\frac{ 2 }{ 5 }&\color{blue}{-4}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}2&-\frac{ 1 }{ 100 }&\frac{ 1 }{ 25 }&\frac{ 7 }{ 20 }&-\frac{ 59 }{ 50 }&-\frac{ 16 }{ 5 }&\color{orangered}{ 8 }\\& & -\frac{ 1 }{ 50 }& \frac{ 1 }{ 25 }& \frac{ 39 }{ 50 }& -\frac{ 4 }{ 5 }& \color{orangered}{-8} \\ \hline &\color{blue}{-\frac{ 1 }{ 100 }}&\color{blue}{\frac{ 1 }{ 50 }}&\color{blue}{\frac{ 39 }{ 100 }}&\color{blue}{-\frac{ 2 }{ 5 }}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -\frac{ 1 }{ 100 }x^{4}+\frac{ 1 }{ 50 }x^{3}+\frac{ 39 }{ 100 }x^{2}-\frac{ 2 }{ 5 }x-4 } $ with a remainder of $ \color{red}{ 0 } $.