Divide using synthetic division $$ ( x^{3}+x^{2}+9x ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&1&1&9&0\\& & 2& 6& \color{black}{30} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{15}&\color{orangered}{30} \end{array} $$The solution is:
$$ \frac{ x^{3}+x^{2}+9x }{ x-2 } = \color{blue}{x^{2}+3x+15} ~+~ \frac{ \color{red}{ 30 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&1&9&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 1 }&1&9&0\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&1&9&0\\& & \color{blue}{2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 2 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}2&1&\color{orangered}{ 1 }&9&0\\& & \color{orangered}{2} & & \\ \hline &1&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&1&9&0\\& & 2& \color{blue}{6} & \\ \hline &1&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 6 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrr}2&1&1&\color{orangered}{ 9 }&0\\& & 2& \color{orangered}{6} & \\ \hline &1&3&\color{orangered}{15}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 15 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&1&9&0\\& & 2& 6& \color{blue}{30} \\ \hline &1&3&\color{blue}{15}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 30 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrr}2&1&1&9&\color{orangered}{ 0 }\\& & 2& 6& \color{orangered}{30} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{15}&\color{orangered}{30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+3x+15 } $ with a remainder of $ \color{red}{ 30 } $.