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

The synthetic division table is:

$$ \begin{array}{c|rrrr}2&1&-3&2&-6\\& & 2& -2& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{0}&\color{orangered}{-6} \end{array} $$

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

We can find remainder using synthetic division method.

Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-3&2&-6\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}2&\color{orangered}{ 1 }&-3&2&-6\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.

$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-3&2&-6\\& & \color{blue}{2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 2 } = \color{orangered}{ -1 } $

$$ \begin{array}{c|rrrr}2&1&\color{orangered}{ -3 }&2&-6\\& & \color{orangered}{2} & & \\ \hline &1&\color{orangered}{-1}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.

$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-3&2&-6\\& & 2& \color{blue}{-2} & \\ \hline &1&\color{blue}{-1}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrr}2&1&-3&\color{orangered}{ 2 }&-6\\& & 2& \color{orangered}{-2} & \\ \hline &1&-1&\color{orangered}{0}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-3&2&-6\\& & 2& -2& \color{blue}{0} \\ \hline &1&-1&\color{blue}{0}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 0 } = \color{orangered}{ -6 } $

$$ \begin{array}{c|rrrr}2&1&-3&2&\color{orangered}{ -6 }\\& & 2& -2& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{0}&\color{orangered}{-6} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -6 }\right) $.

Synthetic Division Calculator