Find remainder when $ x^{3}+x^{2}+9x $ is divided by $ x $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}0&1&1&9&0\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{9}&\color{orangered}{0} \end{array} $$

The remainder when $ x^{3}+x^{2}+9x $ is divided by $ x $ is $ \, \color{red}{ 0 } $.

We can find remainder using synthetic division method.

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

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&1&9&0\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}0&\color{orangered}{ 1 }&1&9&0\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&1&9&0\\& & \color{blue}{0} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $

$$ \begin{array}{c|rrrr}0&1&\color{orangered}{ 1 }&9&0\\& & \color{orangered}{0} & & \\ \hline &1&\color{orangered}{1}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&1&9&0\\& & 0& \color{blue}{0} & \\ \hline &1&\color{blue}{1}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 0 } = \color{orangered}{ 9 } $

$$ \begin{array}{c|rrrr}0&1&1&\color{orangered}{ 9 }&0\\& & 0& \color{orangered}{0} & \\ \hline &1&1&\color{orangered}{9}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 9 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&1&9&0\\& & 0& 0& \color{blue}{0} \\ \hline &1&1&\color{blue}{9}& \end{array} $$

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

$$ \begin{array}{c|rrrr}0&1&1&9&\color{orangered}{ 0 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{9}&\color{orangered}{0} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.

Synthetic Division Calculator