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