Divide using synthetic division $$ ( x^{5}+3x^{4}-x-3 ) \div ( x ) $$

The synthetic division table is:

$$ \begin{array}{c|rrrrrr}0&1&3&0&0&-1&-3\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{-3} \end{array} $$

The solution is:

$$ \frac{ x^{5}+3x^{4}-x-3 }{ x } = \color{blue}{x^{4}+3x^{3}-1} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x } $$

Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&3&0&0&-1&-3\\& & & & & & \\ \hline &&&&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 1 }&3&0&0&-1&-3\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&3&0&0&-1&-3\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $

$$ \begin{array}{c|rrrrrr}0&1&\color{orangered}{ 3 }&0&0&-1&-3\\& & \color{orangered}{0} & & & & \\ \hline &1&\color{orangered}{3}&&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 3 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&3&0&0&-1&-3\\& & 0& \color{blue}{0} & & & \\ \hline &1&\color{blue}{3}&&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrrr}0&1&3&\color{orangered}{ 0 }&0&-1&-3\\& & 0& \color{orangered}{0} & & & \\ \hline &1&3&\color{orangered}{0}&&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&3&0&0&-1&-3\\& & 0& 0& \color{blue}{0} & & \\ \hline &1&3&\color{blue}{0}&&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrrr}0&1&3&0&\color{orangered}{ 0 }&-1&-3\\& & 0& 0& \color{orangered}{0} & & \\ \hline &1&3&0&\color{orangered}{0}&& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&3&0&0&-1&-3\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &1&3&0&\color{blue}{0}&& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 0 } = \color{orangered}{ -1 } $

$$ \begin{array}{c|rrrrrr}0&1&3&0&0&\color{orangered}{ -1 }&-3\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &1&3&0&0&\color{orangered}{-1}& \end{array} $$

Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&3&0&0&-1&-3\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &1&3&0&0&\color{blue}{-1}& \end{array} $$

Step 11 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 0 } = \color{orangered}{ -3 } $

$$ \begin{array}{c|rrrrrr}0&1&3&0&0&-1&\color{orangered}{ -3 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{-3} \end{array} $$

Bottom line represents the quotient $ \color{blue}{ x^{4}+3x^{3}-1 } $ with a remainder of $ \color{red}{ -3 } $.

Synthetic Division Calculator