Divide using synthetic division $$ ( x^{5}-3x^{4}-9x^{3}+27x^{2}+18x-54 ) \div ( 0 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}0&1&-3&-9&27&18&-54\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-9}&\color{blue}{27}&\color{blue}{18}&\color{orangered}{-54} \end{array} $$The solution is:
$$ \frac{ x^{5}-3x^{4}-9x^{3}+27x^{2}+18x-54 }{ x } = \color{blue}{x^{4}-3x^{3}-9x^{2}+27x+18} \color{red}{~-~} \frac{ \color{red}{ 54 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-3&-9&27&18&-54\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 1 }&-3&-9&27&18&-54\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-3&-9&27&18&-54\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 0 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrr}0&1&\color{orangered}{ -3 }&-9&27&18&-54\\& & \color{orangered}{0} & & & & \\ \hline &1&\color{orangered}{-3}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-3&-9&27&18&-54\\& & 0& \color{blue}{0} & & & \\ \hline &1&\color{blue}{-3}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 0 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrr}0&1&-3&\color{orangered}{ -9 }&27&18&-54\\& & 0& \color{orangered}{0} & & & \\ \hline &1&-3&\color{orangered}{-9}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-3&-9&27&18&-54\\& & 0& 0& \color{blue}{0} & & \\ \hline &1&-3&\color{blue}{-9}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ 0 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrrrr}0&1&-3&-9&\color{orangered}{ 27 }&18&-54\\& & 0& 0& \color{orangered}{0} & & \\ \hline &1&-3&-9&\color{orangered}{27}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 27 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-3&-9&27&18&-54\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &1&-3&-9&\color{blue}{27}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 0 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}0&1&-3&-9&27&\color{orangered}{ 18 }&-54\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &1&-3&-9&27&\color{orangered}{18}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 18 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-3&-9&27&18&-54\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &1&-3&-9&27&\color{blue}{18}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 0 } = \color{orangered}{ -54 } $
$$ \begin{array}{c|rrrrrr}0&1&-3&-9&27&18&\color{orangered}{ -54 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-9}&\color{blue}{27}&\color{blue}{18}&\color{orangered}{-54} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-3x^{3}-9x^{2}+27x+18 } $ with a remainder of $ \color{red}{ -54 } $.