Divide using synthetic division $$ ( 3x^{2}+3x-18 ) \div ( 3x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&3&3&-18\\& & 15& \color{black}{90} \\ \hline &\color{blue}{3}&\color{blue}{18}&\color{orangered}{72} \end{array} $$The solution is:
$$ \frac{ 3x^{2}+3x-18 }{ x-5 } = \color{blue}{3x+18} ~+~ \frac{ \color{red}{ 72 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&3&-18\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ 3 }&3&-18\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 3 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&3&-18\\& & \color{blue}{15} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 15 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrr}5&3&\color{orangered}{ 3 }&-18\\& & \color{orangered}{15} & \\ \hline &3&\color{orangered}{18}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 18 } = \color{blue}{ 90 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&3&-18\\& & 15& \color{blue}{90} \\ \hline &3&\color{blue}{18}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 90 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrr}5&3&3&\color{orangered}{ -18 }\\& & 15& \color{orangered}{90} \\ \hline &\color{blue}{3}&\color{blue}{18}&\color{orangered}{72} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x+18 } $ with a remainder of $ \color{red}{ 72 } $.