Divide using synthetic division $$ ( 2x^{4}-3x^{3}-14x^{2}+9x+18 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&2&-3&-14&9&18\\& & 6& 9& -15& \color{black}{-18} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-5}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-3x^{3}-14x^{2}+9x+18 }{ x-3 } = \color{blue}{2x^{3}+3x^{2}-5x-6} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&-3&-14&9&18\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 2 }&-3&-14&9&18\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&-3&-14&9&18\\& & \color{blue}{6} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 6 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}3&2&\color{orangered}{ -3 }&-14&9&18\\& & \color{orangered}{6} & & & \\ \hline &2&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&-3&-14&9&18\\& & 6& \color{blue}{9} & & \\ \hline &2&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 9 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}3&2&-3&\color{orangered}{ -14 }&9&18\\& & 6& \color{orangered}{9} & & \\ \hline &2&3&\color{orangered}{-5}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&-3&-14&9&18\\& & 6& 9& \color{blue}{-15} & \\ \hline &2&3&\color{blue}{-5}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}3&2&-3&-14&\color{orangered}{ 9 }&18\\& & 6& 9& \color{orangered}{-15} & \\ \hline &2&3&-5&\color{orangered}{-6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&-3&-14&9&18\\& & 6& 9& -15& \color{blue}{-18} \\ \hline &2&3&-5&\color{blue}{-6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}3&2&-3&-14&9&\color{orangered}{ 18 }\\& & 6& 9& -15& \color{orangered}{-18} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-5}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+3x^{2}-5x-6 } $ with a remainder of $ \color{red}{ 0 } $.