Divide using synthetic division $$ ( x^{3}-2.5x^{2}-16.5x-220 ) \div ( x-10 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}10&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 10& 75& \color{black}{585} \\ \hline &\color{blue}{1}&\color{blue}{\frac{ 15 }{ 2 }}&\color{blue}{\frac{ 117 }{ 2 }}&\color{orangered}{365} \end{array} $$The solution is:
$$ \frac{ x^{3}-\frac{ 5 }{ 2 }x^{2}-\frac{ 33 }{ 2 }x-220 }{ x-10 } = \color{blue}{x^{2}+\frac{ 15 }{ 2 }x+\frac{ 117 }{ 2 }} ~+~ \frac{ \color{red}{ 365 } }{ x-10 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -10 = 0 $ ( $ x = \color{blue}{ 10 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}10&\color{orangered}{ 1 }&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 1 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & \color{blue}{10} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -\frac{ 5 }{ 2 } } + \color{orangered}{ 10 } = \color{orangered}{ \frac{ 15 }{ 2 } } $
$$ \begin{array}{c|rrrr}10&1&\color{orangered}{ -\frac{ 5 }{ 2 } }&-\frac{ 33 }{ 2 }&-220\\& & \color{orangered}{10} & & \\ \hline &1&\color{orangered}{\frac{ 15 }{ 2 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ \frac{ 15 }{ 2 } } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 10& \color{blue}{75} & \\ \hline &1&\color{blue}{\frac{ 15 }{ 2 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -\frac{ 33 }{ 2 } } + \color{orangered}{ 75 } = \color{orangered}{ \frac{ 117 }{ 2 } } $
$$ \begin{array}{c|rrrr}10&1&-\frac{ 5 }{ 2 }&\color{orangered}{ -\frac{ 33 }{ 2 } }&-220\\& & 10& \color{orangered}{75} & \\ \hline &1&\frac{ 15 }{ 2 }&\color{orangered}{\frac{ 117 }{ 2 }}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ \frac{ 117 }{ 2 } } = \color{blue}{ 585 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 10& 75& \color{blue}{585} \\ \hline &1&\frac{ 15 }{ 2 }&\color{blue}{\frac{ 117 }{ 2 }}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -220 } + \color{orangered}{ 585 } = \color{orangered}{ 365 } $
$$ \begin{array}{c|rrrr}10&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&\color{orangered}{ -220 }\\& & 10& 75& \color{orangered}{585} \\ \hline &\color{blue}{1}&\color{blue}{\frac{ 15 }{ 2 }}&\color{blue}{\frac{ 117 }{ 2 }}&\color{orangered}{365} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+\frac{ 15 }{ 2 }x+\frac{ 117 }{ 2 } } $ with a remainder of $ \color{red}{ 365 } $.