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

The synthetic division table is:

$$ \begin{array}{c|rrrr}2&1&1&-2&-2\\& & 2& 6& \color{black}{8} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{4}&\color{orangered}{6} \end{array} $$

The remainder when $ x^{3}+x^{2}-2x-2 $ 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.

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

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

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

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 } $.

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

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

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

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.

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

Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 6 } = \color{orangered}{ 4 } $

$$ \begin{array}{c|rrrr}2&1&1&\color{orangered}{ -2 }&-2\\& & 2& \color{orangered}{6} & \\ \hline &1&3&\color{orangered}{4}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.

$$ \begin{array}{c|rrrr}\color{blue}{2}&1&1&-2&-2\\& & 2& 6& \color{blue}{8} \\ \hline &1&3&\color{blue}{4}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 8 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrrr}2&1&1&-2&\color{orangered}{ -2 }\\& & 2& 6& \color{orangered}{8} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{4}&\color{orangered}{6} \end{array} $$

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

Synthetic Division Calculator