Divide using synthetic division $$ ( 3x^{3}+x^{2}-24x-18 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&3&1&-24&-18\\& & 9& 30& \color{black}{18} \\ \hline &\color{blue}{3}&\color{blue}{10}&\color{blue}{6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+x^{2}-24x-18 }{ x-3 } = \color{blue}{3x^{2}+10x+6} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&1&-24&-18\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 3 }&1&-24&-18\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&1&-24&-18\\& & \color{blue}{9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 9 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}3&3&\color{orangered}{ 1 }&-24&-18\\& & \color{orangered}{9} & & \\ \hline &3&\color{orangered}{10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 10 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&1&-24&-18\\& & 9& \color{blue}{30} & \\ \hline &3&\color{blue}{10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 30 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}3&3&1&\color{orangered}{ -24 }&-18\\& & 9& \color{orangered}{30} & \\ \hline &3&10&\color{orangered}{6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&1&-24&-18\\& & 9& 30& \color{blue}{18} \\ \hline &3&10&\color{blue}{6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 18 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}3&3&1&-24&\color{orangered}{ -18 }\\& & 9& 30& \color{orangered}{18} \\ \hline &\color{blue}{3}&\color{blue}{10}&\color{blue}{6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+10x+6 } $ with a remainder of $ \color{red}{ 0 } $.