Find remainder when $ x^{4}-x^{3}+2x^{2}-x-1 $ is divided by $ x-1 $.

The synthetic division table is:

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

The remainder when $ x^{4}-x^{3}+2x^{2}-x-1 $ is divided by $ x-1 $ is $ \, \color{red}{ 0 } $.

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 -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.

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

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

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

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

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

Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 1 } = \color{orangered}{ 0 } $

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

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

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

Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $

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

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.

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

Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 2 } = \color{orangered}{ 1 } $

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

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.

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

Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 1 } = \color{orangered}{ 0 } $

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

Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.

Synthetic Division Calculator