Find remainder when $ x^{3}+12x^{2}+23x-36 $ is divided by $ -4 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}0&1&12&23&-36\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{blue}{23}&\color{orangered}{-36} \end{array} $$

The remainder when $ x^{3}+12x^{2}+23x-36 $ is divided by $ x $ is $ \, \color{red}{ -36 } $.

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.

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&12&23&-36\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}0&\color{orangered}{ 1 }&12&23&-36\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&12&23&-36\\& & \color{blue}{0} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 0 } = \color{orangered}{ 12 } $

$$ \begin{array}{c|rrrr}0&1&\color{orangered}{ 12 }&23&-36\\& & \color{orangered}{0} & & \\ \hline &1&\color{orangered}{12}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 12 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&12&23&-36\\& & 0& \color{blue}{0} & \\ \hline &1&\color{blue}{12}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ 0 } = \color{orangered}{ 23 } $

$$ \begin{array}{c|rrrr}0&1&12&\color{orangered}{ 23 }&-36\\& & 0& \color{orangered}{0} & \\ \hline &1&12&\color{orangered}{23}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 23 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&1&12&23&-36\\& & 0& 0& \color{blue}{0} \\ \hline &1&12&\color{blue}{23}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 0 } = \color{orangered}{ -36 } $

$$ \begin{array}{c|rrrr}0&1&12&23&\color{orangered}{ -36 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{blue}{23}&\color{orangered}{-36} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -36 }\right) $.

Synthetic Division Calculator