Divide using synthetic division $$ ( 0.1x^{5}-0.9x^{4}+2.7x^{3}-3.1x^{2}+1.2x ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & -\frac{ 2 }{ 5 }& \frac{ 26 }{ 5 }& -\frac{ 158 }{ 5 }& \frac{ 694 }{ 5 }& \color{black}{-560} \\ \hline &\color{blue}{\frac{ 1 }{ 10 }}&\color{blue}{-\frac{ 13 }{ 10 }}&\color{blue}{\frac{ 79 }{ 10 }}&\color{blue}{-\frac{ 347 }{ 10 }}&\color{blue}{140}&\color{orangered}{-560} \end{array} $$The solution is:
$$ \frac{ \frac{ 1 }{ 10 }x^{5}-\frac{ 9 }{ 10 }x^{4}+\frac{ 27 }{ 10 }x^{3}-\frac{ 31 }{ 10 }x^{2}+\frac{ 6 }{ 5 }x }{ x+4 } = \color{blue}{\frac{ 1 }{ 10 }x^{4}-\frac{ 13 }{ 10 }x^{3}+\frac{ 79 }{ 10 }x^{2}-\frac{ 347 }{ 10 }x+140} \color{red}{~-~} \frac{ \color{red}{ 560 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ \frac{ 1 }{ 10 } }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & & & & & \\ \hline &\color{orangered}{\frac{ 1 }{ 10 }}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \frac{ 1 }{ 10 } } = \color{blue}{ -\frac{ 2 }{ 5 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & \color{blue}{-\frac{ 2 }{ 5 }} & & & & \\ \hline &\color{blue}{\frac{ 1 }{ 10 }}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -\frac{ 9 }{ 10 } } + \color{orangered}{ \left( -\frac{ 2 }{ 5 } \right) } = \color{orangered}{ -\frac{ 13 }{ 10 } } $
$$ \begin{array}{c|rrrrrr}-4&\frac{ 1 }{ 10 }&\color{orangered}{ -\frac{ 9 }{ 10 } }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & \color{orangered}{-\frac{ 2 }{ 5 }} & & & & \\ \hline &\frac{ 1 }{ 10 }&\color{orangered}{-\frac{ 13 }{ 10 }}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -\frac{ 13 }{ 10 } \right) } = \color{blue}{ \frac{ 26 }{ 5 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & -\frac{ 2 }{ 5 }& \color{blue}{\frac{ 26 }{ 5 }} & & & \\ \hline &\frac{ 1 }{ 10 }&\color{blue}{-\frac{ 13 }{ 10 }}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ \frac{ 27 }{ 10 } } + \color{orangered}{ \frac{ 26 }{ 5 } } = \color{orangered}{ \frac{ 79 }{ 10 } } $
$$ \begin{array}{c|rrrrrr}-4&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\color{orangered}{ \frac{ 27 }{ 10 } }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & -\frac{ 2 }{ 5 }& \color{orangered}{\frac{ 26 }{ 5 }} & & & \\ \hline &\frac{ 1 }{ 10 }&-\frac{ 13 }{ 10 }&\color{orangered}{\frac{ 79 }{ 10 }}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \frac{ 79 }{ 10 } } = \color{blue}{ -\frac{ 158 }{ 5 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & -\frac{ 2 }{ 5 }& \frac{ 26 }{ 5 }& \color{blue}{-\frac{ 158 }{ 5 }} & & \\ \hline &\frac{ 1 }{ 10 }&-\frac{ 13 }{ 10 }&\color{blue}{\frac{ 79 }{ 10 }}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -\frac{ 31 }{ 10 } } + \color{orangered}{ \left( -\frac{ 158 }{ 5 } \right) } = \color{orangered}{ -\frac{ 347 }{ 10 } } $
$$ \begin{array}{c|rrrrrr}-4&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&\color{orangered}{ -\frac{ 31 }{ 10 } }&\frac{ 6 }{ 5 }&0\\& & -\frac{ 2 }{ 5 }& \frac{ 26 }{ 5 }& \color{orangered}{-\frac{ 158 }{ 5 }} & & \\ \hline &\frac{ 1 }{ 10 }&-\frac{ 13 }{ 10 }&\frac{ 79 }{ 10 }&\color{orangered}{-\frac{ 347 }{ 10 }}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -\frac{ 347 }{ 10 } \right) } = \color{blue}{ \frac{ 694 }{ 5 } } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & -\frac{ 2 }{ 5 }& \frac{ 26 }{ 5 }& -\frac{ 158 }{ 5 }& \color{blue}{\frac{ 694 }{ 5 }} & \\ \hline &\frac{ 1 }{ 10 }&-\frac{ 13 }{ 10 }&\frac{ 79 }{ 10 }&\color{blue}{-\frac{ 347 }{ 10 }}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ \frac{ 6 }{ 5 } } + \color{orangered}{ \frac{ 694 }{ 5 } } = \color{orangered}{ 140 } $
$$ \begin{array}{c|rrrrrr}-4&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\color{orangered}{ \frac{ 6 }{ 5 } }&0\\& & -\frac{ 2 }{ 5 }& \frac{ 26 }{ 5 }& -\frac{ 158 }{ 5 }& \color{orangered}{\frac{ 694 }{ 5 }} & \\ \hline &\frac{ 1 }{ 10 }&-\frac{ 13 }{ 10 }&\frac{ 79 }{ 10 }&-\frac{ 347 }{ 10 }&\color{orangered}{140}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 140 } = \color{blue}{ -560 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&0\\& & -\frac{ 2 }{ 5 }& \frac{ 26 }{ 5 }& -\frac{ 158 }{ 5 }& \frac{ 694 }{ 5 }& \color{blue}{-560} \\ \hline &\frac{ 1 }{ 10 }&-\frac{ 13 }{ 10 }&\frac{ 79 }{ 10 }&-\frac{ 347 }{ 10 }&\color{blue}{140}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -560 \right) } = \color{orangered}{ -560 } $
$$ \begin{array}{c|rrrrrr}-4&\frac{ 1 }{ 10 }&-\frac{ 9 }{ 10 }&\frac{ 27 }{ 10 }&-\frac{ 31 }{ 10 }&\frac{ 6 }{ 5 }&\color{orangered}{ 0 }\\& & -\frac{ 2 }{ 5 }& \frac{ 26 }{ 5 }& -\frac{ 158 }{ 5 }& \frac{ 694 }{ 5 }& \color{orangered}{-560} \\ \hline &\color{blue}{\frac{ 1 }{ 10 }}&\color{blue}{-\frac{ 13 }{ 10 }}&\color{blue}{\frac{ 79 }{ 10 }}&\color{blue}{-\frac{ 347 }{ 10 }}&\color{blue}{140}&\color{orangered}{-560} \end{array} $$Bottom line represents the quotient $ \color{blue}{ \frac{ 1 }{ 10 }x^{4}-\frac{ 13 }{ 10 }x^{3}+\frac{ 79 }{ 10 }x^{2}-\frac{ 347 }{ 10 }x+140 } $ with a remainder of $ \color{red}{ -560 } $.