Divide using synthetic division $$ ( x^{3}-2.5x^{2}-16.5x-220 ) \div ( x-20 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}20&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 20& 350& \color{black}{6670} \\ \hline &\color{blue}{1}&\color{blue}{\frac{ 35 }{ 2 }}&\color{blue}{\frac{ 667 }{ 2 }}&\color{orangered}{6450} \end{array} $$The solution is:
$$ \frac{ x^{3}-\frac{ 5 }{ 2 }x^{2}-\frac{ 33 }{ 2 }x-220 }{ x-20 } = \color{blue}{x^{2}+\frac{ 35 }{ 2 }x+\frac{ 667 }{ 2 }} ~+~ \frac{ \color{red}{ 6450 } }{ x-20 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -20 = 0 $ ( $ x = \color{blue}{ 20 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}20&\color{orangered}{ 1 }&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 1 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & \color{blue}{20} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -\frac{ 5 }{ 2 } } + \color{orangered}{ 20 } = \color{orangered}{ \frac{ 35 }{ 2 } } $
$$ \begin{array}{c|rrrr}20&1&\color{orangered}{ -\frac{ 5 }{ 2 } }&-\frac{ 33 }{ 2 }&-220\\& & \color{orangered}{20} & & \\ \hline &1&\color{orangered}{\frac{ 35 }{ 2 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ \frac{ 35 }{ 2 } } = \color{blue}{ 350 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 20& \color{blue}{350} & \\ \hline &1&\color{blue}{\frac{ 35 }{ 2 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -\frac{ 33 }{ 2 } } + \color{orangered}{ 350 } = \color{orangered}{ \frac{ 667 }{ 2 } } $
$$ \begin{array}{c|rrrr}20&1&-\frac{ 5 }{ 2 }&\color{orangered}{ -\frac{ 33 }{ 2 } }&-220\\& & 20& \color{orangered}{350} & \\ \hline &1&\frac{ 35 }{ 2 }&\color{orangered}{\frac{ 667 }{ 2 }}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ \frac{ 667 }{ 2 } } = \color{blue}{ 6670 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&-220\\& & 20& 350& \color{blue}{6670} \\ \hline &1&\frac{ 35 }{ 2 }&\color{blue}{\frac{ 667 }{ 2 }}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -220 } + \color{orangered}{ 6670 } = \color{orangered}{ 6450 } $
$$ \begin{array}{c|rrrr}20&1&-\frac{ 5 }{ 2 }&-\frac{ 33 }{ 2 }&\color{orangered}{ -220 }\\& & 20& 350& \color{orangered}{6670} \\ \hline &\color{blue}{1}&\color{blue}{\frac{ 35 }{ 2 }}&\color{blue}{\frac{ 667 }{ 2 }}&\color{orangered}{6450} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+\frac{ 35 }{ 2 }x+\frac{ 667 }{ 2 } } $ with a remainder of $ \color{red}{ 6450 } $.