Divide using synthetic division $$ ( x^{3}-2.5x^{2}-16.5x-220 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 4& 6& \color{black}{-42} \\ \hline &\color{blue}{1}&\color{blue}{\frac{ 3 }{ 2 }}&\color{blue}{-\frac{ 21 }{ 2 }}&\color{orangered}{-262} \end{array} $$The solution is:
$$ \frac{ x^{3}-\frac{ 5 }{ 2 }x^{2}-\frac{ 33 }{ 2 }x-220 }{ x-4 } = \color{blue}{x^{2}+\frac{ 3 }{ 2 }x-\frac{ 21 }{ 2 }} \color{red}{~-~} \frac{ \color{red}{ 262 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 1 }&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & \color{blue}{4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -\frac{ 5 }{ 2 } } + \color{orangered}{ 4 } = \color{orangered}{ \frac{ 3 }{ 2 } } $
$$ \begin{array}{c|rrrr}4&1&\color{orangered}{ -\frac{ 5 }{ 2 } }&-\frac{ 33 }{ 2 }&-220\\& & \color{orangered}{4} & & \\ \hline &1&\color{orangered}{\frac{ 3 }{ 2 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \frac{ 3 }{ 2 } } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 4& \color{blue}{6} & \\ \hline &1&\color{blue}{\frac{ 3 }{ 2 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -\frac{ 33 }{ 2 } } + \color{orangered}{ 6 } = \color{orangered}{ -\frac{ 21 }{ 2 } } $
$$ \begin{array}{c|rrrr}4&1&-\frac{ 5 }{ 2 }&\color{orangered}{ -\frac{ 33 }{ 2 } }&-220\\& & 4& \color{orangered}{6} & \\ \hline &1&\frac{ 3 }{ 2 }&\color{orangered}{-\frac{ 21 }{ 2 }}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -\frac{ 21 }{ 2 } \right) } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 4& 6& \color{blue}{-42} \\ \hline &1&\frac{ 3 }{ 2 }&\color{blue}{-\frac{ 21 }{ 2 }}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -220 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -262 } $
$$ \begin{array}{c|rrrr}4&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&\color{orangered}{ -220 }\\& & 4& 6& \color{orangered}{-42} \\ \hline &\color{blue}{1}&\color{blue}{\frac{ 3 }{ 2 }}&\color{blue}{-\frac{ 21 }{ 2 }}&\color{orangered}{-262} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+\frac{ 3 }{ 2 }x-\frac{ 21 }{ 2 } } $ with a remainder of $ \color{red}{ -262 } $.