Find remainder when $ 3x^{3}-7x^{2}+6x-14 $ is divided by $ x $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}0&3&-7&6&-14\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{-7}&\color{blue}{6}&\color{orangered}{-14} \end{array} $$

The remainder when $ 3x^{3}-7x^{2}+6x-14 $ is divided by $ x $ is $ \, \color{red}{ -14 } $.

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.

$$ \begin{array}{c|rrrr}\color{blue}{0}&3&-7&6&-14\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}0&\color{orangered}{ 3 }&-7&6&-14\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&3&-7&6&-14\\& & \color{blue}{0} & & \\ \hline &\color{blue}{3}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 0 } = \color{orangered}{ -7 } $

$$ \begin{array}{c|rrrr}0&3&\color{orangered}{ -7 }&6&-14\\& & \color{orangered}{0} & & \\ \hline &3&\color{orangered}{-7}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{0}&3&-7&6&-14\\& & 0& \color{blue}{0} & \\ \hline &3&\color{blue}{-7}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 0 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrrr}0&3&-7&\color{orangered}{ 6 }&-14\\& & 0& \color{orangered}{0} & \\ \hline &3&-7&\color{orangered}{6}& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{0}&3&-7&6&-14\\& & 0& 0& \color{blue}{0} \\ \hline &3&-7&\color{blue}{6}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 0 } = \color{orangered}{ -14 } $

$$ \begin{array}{c|rrrr}0&3&-7&6&\color{orangered}{ -14 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{-7}&\color{blue}{6}&\color{orangered}{-14} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -14 }\right) $.

Synthetic Division Calculator