Divide using synthetic division $$ ( 0.4x^{3}-0.018x^{2}+0.11x+0.06 ) \div ( x-0.3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 10 }&\frac{ 2 }{ 5 }&-\frac{ 9 }{ 500 }&\frac{ 11 }{ 100 }&\frac{ 3 }{ 50 }\\& & \frac{ 3 }{ 25 }& \frac{ 153 }{ 5000 }& \color{black}{\frac{ 2109 }{ 50000 }} \\ \hline &\color{blue}{\frac{ 2 }{ 5 }}&\color{blue}{\frac{ 51 }{ 500 }}&\color{blue}{\frac{ 703 }{ 5000 }}&\color{orangered}{\frac{ 5109 }{ 50000 }} \end{array} $$The solution is:
$$ \frac{ \frac{ 2 }{ 5 }x^{3}-\frac{ 9 }{ 500 }x^{2}+\frac{ 11 }{ 100 }x+\frac{ 3 }{ 50 } }{ x-\frac{ 3 }{ 10 } } = \color{blue}{\frac{ 2 }{ 5 }x^{2}+\frac{ 51 }{ 500 }x+\frac{ 703 }{ 5000 }} ~+~ \frac{ \color{red}{ \frac{ 5109 }{ 50000 } } }{ x-\frac{ 3 }{ 10 } } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 3 }{ 10 } = 0 $ ( $ x = \color{blue}{ \frac{ 3 }{ 10 } } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 10 }}&\frac{ 2 }{ 5 }&-\frac{ 9 }{ 500 }&\frac{ 11 }{ 100 }&\frac{ 3 }{ 50 }\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 10 }&\color{orangered}{ \frac{ 2 }{ 5 } }&-\frac{ 9 }{ 500 }&\frac{ 11 }{ 100 }&\frac{ 3 }{ 50 }\\& & & & \\ \hline &\color{orangered}{\frac{ 2 }{ 5 }}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 3 }{ 10 } } \cdot \color{blue}{ \frac{ 2 }{ 5 } } = \color{blue}{ \frac{ 3 }{ 25 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 10 }}&\frac{ 2 }{ 5 }&-\frac{ 9 }{ 500 }&\frac{ 11 }{ 100 }&\frac{ 3 }{ 50 }\\& & \color{blue}{\frac{ 3 }{ 25 }} & & \\ \hline &\color{blue}{\frac{ 2 }{ 5 }}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -\frac{ 9 }{ 500 } } + \color{orangered}{ \frac{ 3 }{ 25 } } = \color{orangered}{ \frac{ 51 }{ 500 } } $
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 10 }&\frac{ 2 }{ 5 }&\color{orangered}{ -\frac{ 9 }{ 500 } }&\frac{ 11 }{ 100 }&\frac{ 3 }{ 50 }\\& & \color{orangered}{\frac{ 3 }{ 25 }} & & \\ \hline &\frac{ 2 }{ 5 }&\color{orangered}{\frac{ 51 }{ 500 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 3 }{ 10 } } \cdot \color{blue}{ \frac{ 51 }{ 500 } } = \color{blue}{ \frac{ 153 }{ 5000 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 10 }}&\frac{ 2 }{ 5 }&-\frac{ 9 }{ 500 }&\frac{ 11 }{ 100 }&\frac{ 3 }{ 50 }\\& & \frac{ 3 }{ 25 }& \color{blue}{\frac{ 153 }{ 5000 }} & \\ \hline &\frac{ 2 }{ 5 }&\color{blue}{\frac{ 51 }{ 500 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ \frac{ 11 }{ 100 } } + \color{orangered}{ \frac{ 153 }{ 5000 } } = \color{orangered}{ \frac{ 703 }{ 5000 } } $
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 10 }&\frac{ 2 }{ 5 }&-\frac{ 9 }{ 500 }&\color{orangered}{ \frac{ 11 }{ 100 } }&\frac{ 3 }{ 50 }\\& & \frac{ 3 }{ 25 }& \color{orangered}{\frac{ 153 }{ 5000 }} & \\ \hline &\frac{ 2 }{ 5 }&\frac{ 51 }{ 500 }&\color{orangered}{\frac{ 703 }{ 5000 }}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 3 }{ 10 } } \cdot \color{blue}{ \frac{ 703 }{ 5000 } } = \color{blue}{ \frac{ 2109 }{ 50000 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 3 }{ 10 }}&\frac{ 2 }{ 5 }&-\frac{ 9 }{ 500 }&\frac{ 11 }{ 100 }&\frac{ 3 }{ 50 }\\& & \frac{ 3 }{ 25 }& \frac{ 153 }{ 5000 }& \color{blue}{\frac{ 2109 }{ 50000 }} \\ \hline &\frac{ 2 }{ 5 }&\frac{ 51 }{ 500 }&\color{blue}{\frac{ 703 }{ 5000 }}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ \frac{ 3 }{ 50 } } + \color{orangered}{ \frac{ 2109 }{ 50000 } } = \color{orangered}{ \frac{ 5109 }{ 50000 } } $
$$ \begin{array}{c|rrrr}\frac{ 3 }{ 10 }&\frac{ 2 }{ 5 }&-\frac{ 9 }{ 500 }&\frac{ 11 }{ 100 }&\color{orangered}{ \frac{ 3 }{ 50 } }\\& & \frac{ 3 }{ 25 }& \frac{ 153 }{ 5000 }& \color{orangered}{\frac{ 2109 }{ 50000 }} \\ \hline &\color{blue}{\frac{ 2 }{ 5 }}&\color{blue}{\frac{ 51 }{ 500 }}&\color{blue}{\frac{ 703 }{ 5000 }}&\color{orangered}{\frac{ 5109 }{ 50000 }} \end{array} $$Bottom line represents the quotient $ \color{blue}{ \frac{ 2 }{ 5 }x^{2}+\frac{ 51 }{ 500 }x+\frac{ 703 }{ 5000 } } $ with a remainder of $ \color{red}{ \frac{ 5109 }{ 50000 } } $.