Divide using synthetic division $$ ( 3x^{3}+18x^{2}+7x-6 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&3&18&7&-6\\& & -9& -27& \color{black}{60} \\ \hline &\color{blue}{3}&\color{blue}{9}&\color{blue}{-20}&\color{orangered}{54} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+18x^{2}+7x-6 }{ x+3 } = \color{blue}{3x^{2}+9x-20} ~+~ \frac{ \color{red}{ 54 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&18&7&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 3 }&18&7&-6\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&18&7&-6\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}-3&3&\color{orangered}{ 18 }&7&-6\\& & \color{orangered}{-9} & & \\ \hline &3&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 9 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&18&7&-6\\& & -9& \color{blue}{-27} & \\ \hline &3&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrr}-3&3&18&\color{orangered}{ 7 }&-6\\& & -9& \color{orangered}{-27} & \\ \hline &3&9&\color{orangered}{-20}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&18&7&-6\\& & -9& -27& \color{blue}{60} \\ \hline &3&9&\color{blue}{-20}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 60 } = \color{orangered}{ 54 } $
$$ \begin{array}{c|rrrr}-3&3&18&7&\color{orangered}{ -6 }\\& & -9& -27& \color{orangered}{60} \\ \hline &\color{blue}{3}&\color{blue}{9}&\color{blue}{-20}&\color{orangered}{54} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+9x-20 } $ with a remainder of $ \color{red}{ 54 } $.