Find remainder when $ 3x^{2}+6x+23 $ is divided by $ 2 $.

The synthetic division table is:

$$ \begin{array}{c|rrr}0&3&6&23\\& & 0& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{orangered}{23} \end{array} $$

The remainder when $ 3x^{2}+6x+23 $ is divided by $ x $ is $ \, \color{red}{ 23 } $.

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.

$$ \begin{array}{c|rrr}\color{blue}{0}&3&6&23\\& & & \\ \hline &&& \end{array} $$

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

$$ \begin{array}{c|rrr}0&\color{orangered}{ 3 }&6&23\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrr}\color{blue}{0}&3&6&23\\& & \color{blue}{0} & \\ \hline &\color{blue}{3}&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 0 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrr}0&3&\color{orangered}{ 6 }&23\\& & \color{orangered}{0} & \\ \hline &3&\color{orangered}{6}& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 6 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrr}\color{blue}{0}&3&6&23\\& & 0& \color{blue}{0} \\ \hline &3&\color{blue}{6}& \end{array} $$

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

$$ \begin{array}{c|rrr}0&3&6&\color{orangered}{ 23 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{orangered}{23} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 23 }\right) $.

Synthetic Division Calculator