Divide using synthetic division $$ ( x^{3}-2.5x^{2}-16.5x-220 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & -2& 9& \color{black}{15} \\ \hline &\color{blue}{1}&\color{blue}{-\frac{ 9 }{ 2 }}&\color{blue}{-\frac{ 15 }{ 2 }}&\color{orangered}{-205} \end{array} $$The solution is:
$$ \frac{ x^{3}-\frac{ 5 }{ 2 }x^{2}-\frac{ 33 }{ 2 }x-220 }{ x+2 } = \color{blue}{x^{2}-\frac{ 9 }{ 2 }x-\frac{ 15 }{ 2 }} \color{red}{~-~} \frac{ \color{red}{ 205 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 1 }&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -\frac{ 5 }{ 2 } } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -\frac{ 9 }{ 2 } } $
$$ \begin{array}{c|rrrr}-2&1&\color{orangered}{ -\frac{ 5 }{ 2 } }&-\frac{ 33 }{ 2 }&-220\\& & \color{orangered}{-2} & & \\ \hline &1&\color{orangered}{-\frac{ 9 }{ 2 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -\frac{ 9 }{ 2 } \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & -2& \color{blue}{9} & \\ \hline &1&\color{blue}{-\frac{ 9 }{ 2 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -\frac{ 33 }{ 2 } } + \color{orangered}{ 9 } = \color{orangered}{ -\frac{ 15 }{ 2 } } $
$$ \begin{array}{c|rrrr}-2&1&-\frac{ 5 }{ 2 }&\color{orangered}{ -\frac{ 33 }{ 2 } }&-220\\& & -2& \color{orangered}{9} & \\ \hline &1&-\frac{ 9 }{ 2 }&\color{orangered}{-\frac{ 15 }{ 2 }}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -\frac{ 15 }{ 2 } \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & -2& 9& \color{blue}{15} \\ \hline &1&-\frac{ 9 }{ 2 }&\color{blue}{-\frac{ 15 }{ 2 }}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -220 } + \color{orangered}{ 15 } = \color{orangered}{ -205 } $
$$ \begin{array}{c|rrrr}-2&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&\color{orangered}{ -220 }\\& & -2& 9& \color{orangered}{15} \\ \hline &\color{blue}{1}&\color{blue}{-\frac{ 9 }{ 2 }}&\color{blue}{-\frac{ 15 }{ 2 }}&\color{orangered}{-205} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-\frac{ 9 }{ 2 }x-\frac{ 15 }{ 2 } } $ with a remainder of $ \color{red}{ -205 } $.