Divide using synthetic division $$ ( 3x^{4}-2x^{3}+26x^{2}-18x-9 ) \div ( 3x ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&3&-2&26&-18&-9\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{blue}{26}&\color{blue}{-18}&\color{orangered}{-9} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-2x^{3}+26x^{2}-18x-9 }{ x } = \color{blue}{3x^{3}-2x^{2}+26x-18} \color{red}{~-~} \frac{ \color{red}{ 9 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&26&-18&-9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 3 }&-2&26&-18&-9\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 3 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&26&-18&-9\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 0 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}0&3&\color{orangered}{ -2 }&26&-18&-9\\& & \color{orangered}{0} & & & \\ \hline &3&\color{orangered}{-2}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&26&-18&-9\\& & 0& \color{blue}{0} & & \\ \hline &3&\color{blue}{-2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ 0 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrrr}0&3&-2&\color{orangered}{ 26 }&-18&-9\\& & 0& \color{orangered}{0} & & \\ \hline &3&-2&\color{orangered}{26}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 26 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&26&-18&-9\\& & 0& 0& \color{blue}{0} & \\ \hline &3&-2&\color{blue}{26}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 0 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}0&3&-2&26&\color{orangered}{ -18 }&-9\\& & 0& 0& \color{orangered}{0} & \\ \hline &3&-2&26&\color{orangered}{-18}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&26&-18&-9\\& & 0& 0& 0& \color{blue}{0} \\ \hline &3&-2&26&\color{blue}{-18}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 0 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}0&3&-2&26&-18&\color{orangered}{ -9 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{blue}{26}&\color{blue}{-18}&\color{orangered}{-9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-2x^{2}+26x-18 } $ with a remainder of $ \color{red}{ -9 } $.