Divide using synthetic division $$ ( 2x^{3}+3x^{2}+5x+9 ) \div ( x-20 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}20&2&3&5&9\\& & 40& 860& \color{black}{17300} \\ \hline &\color{blue}{2}&\color{blue}{43}&\color{blue}{865}&\color{orangered}{17309} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+3x^{2}+5x+9 }{ x-20 } = \color{blue}{2x^{2}+43x+865} ~+~ \frac{ \color{red}{ 17309 } }{ x-20 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{20}&2&3&5&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}20&\color{orangered}{ 2 }&3&5&9\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 2 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&2&3&5&9\\& & \color{blue}{40} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 40 } = \color{orangered}{ 43 } $
$$ \begin{array}{c|rrrr}20&2&\color{orangered}{ 3 }&5&9\\& & \color{orangered}{40} & & \\ \hline &2&\color{orangered}{43}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 43 } = \color{blue}{ 860 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&2&3&5&9\\& & 40& \color{blue}{860} & \\ \hline &2&\color{blue}{43}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 860 } = \color{orangered}{ 865 } $
$$ \begin{array}{c|rrrr}20&2&3&\color{orangered}{ 5 }&9\\& & 40& \color{orangered}{860} & \\ \hline &2&43&\color{orangered}{865}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 865 } = \color{blue}{ 17300 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&2&3&5&9\\& & 40& 860& \color{blue}{17300} \\ \hline &2&43&\color{blue}{865}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 17300 } = \color{orangered}{ 17309 } $
$$ \begin{array}{c|rrrr}20&2&3&5&\color{orangered}{ 9 }\\& & 40& 860& \color{orangered}{17300} \\ \hline &\color{blue}{2}&\color{blue}{43}&\color{blue}{865}&\color{orangered}{17309} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+43x+865 } $ with a remainder of $ \color{red}{ 17309 } $.