Divide using synthetic division $$ ( -0.4x^{3}+128x^{2}-1800x-180000 ) \div ( x-50 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}50&-\frac{ 2 }{ 5 }&128&-1800&-180000\\& & -20& 5400& \color{black}{180000} \\ \hline &\color{blue}{-\frac{ 2 }{ 5 }}&\color{blue}{108}&\color{blue}{3600}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -\frac{ 2 }{ 5 }x^{3}+128x^{2}-1800x-180000 }{ x-50 } = \color{blue}{-\frac{ 2 }{ 5 }x^{2}+108x+3600} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -50 = 0 $ ( $ x = \color{blue}{ 50 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{50}&-\frac{ 2 }{ 5 }&128&-1800&-180000\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}50&\color{orangered}{ -\frac{ 2 }{ 5 } }&128&-1800&-180000\\& & & & \\ \hline &\color{orangered}{-\frac{ 2 }{ 5 }}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 50 } \cdot \color{blue}{ \left( -\frac{ 2 }{ 5 } \right) } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{50}&-\frac{ 2 }{ 5 }&128&-1800&-180000\\& & \color{blue}{-20} & & \\ \hline &\color{blue}{-\frac{ 2 }{ 5 }}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 128 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrrr}50&-\frac{ 2 }{ 5 }&\color{orangered}{ 128 }&-1800&-180000\\& & \color{orangered}{-20} & & \\ \hline &-\frac{ 2 }{ 5 }&\color{orangered}{108}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 50 } \cdot \color{blue}{ 108 } = \color{blue}{ 5400 } $.
$$ \begin{array}{c|rrrr}\color{blue}{50}&-\frac{ 2 }{ 5 }&128&-1800&-180000\\& & -20& \color{blue}{5400} & \\ \hline &-\frac{ 2 }{ 5 }&\color{blue}{108}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1800 } + \color{orangered}{ 5400 } = \color{orangered}{ 3600 } $
$$ \begin{array}{c|rrrr}50&-\frac{ 2 }{ 5 }&128&\color{orangered}{ -1800 }&-180000\\& & -20& \color{orangered}{5400} & \\ \hline &-\frac{ 2 }{ 5 }&108&\color{orangered}{3600}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 50 } \cdot \color{blue}{ 3600 } = \color{blue}{ 180000 } $.
$$ \begin{array}{c|rrrr}\color{blue}{50}&-\frac{ 2 }{ 5 }&128&-1800&-180000\\& & -20& 5400& \color{blue}{180000} \\ \hline &-\frac{ 2 }{ 5 }&108&\color{blue}{3600}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -180000 } + \color{orangered}{ 180000 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}50&-\frac{ 2 }{ 5 }&128&-1800&\color{orangered}{ -180000 }\\& & -20& 5400& \color{orangered}{180000} \\ \hline &\color{blue}{-\frac{ 2 }{ 5 }}&\color{blue}{108}&\color{blue}{3600}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -\frac{ 2 }{ 5 }x^{2}+108x+3600 } $ with a remainder of $ \color{red}{ 0 } $.