Find remainder when $ x^{5}+5x^{2}+x-7 $ is divided by $ x-1 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrrr}1&1&0&0&5&1&-7\\& & 1& 1& 1& 6& \color{black}{7} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{blue}{7}&\color{orangered}{0} \end{array} $$

The remainder when $ x^{5}+5x^{2}+x-7 $ 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.

$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&0&0&5&1&-7\\& & & & & & \\ \hline &&&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}1&\color{orangered}{ 1 }&0&0&5&1&-7\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&0&0&5&1&-7\\& & \color{blue}{1} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}1&1&\color{orangered}{ 0 }&0&5&1&-7\\& & \color{orangered}{1} & & & & \\ \hline &1&\color{orangered}{1}&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&0&0&5&1&-7\\& & 1& \color{blue}{1} & & & \\ \hline &1&\color{blue}{1}&&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1 } = \color{orangered}{ 1 } $

$$ \begin{array}{c|rrrrrr}1&1&0&\color{orangered}{ 0 }&5&1&-7\\& & 1& \color{orangered}{1} & & & \\ \hline &1&1&\color{orangered}{1}&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&0&0&5&1&-7\\& & 1& 1& \color{blue}{1} & & \\ \hline &1&1&\color{blue}{1}&&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 1 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrrrrr}1&1&0&0&\color{orangered}{ 5 }&1&-7\\& & 1& 1& \color{orangered}{1} & & \\ \hline &1&1&1&\color{orangered}{6}&& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&0&0&5&1&-7\\& & 1& 1& 1& \color{blue}{6} & \\ \hline &1&1&1&\color{blue}{6}&& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 6 } = \color{orangered}{ 7 } $

$$ \begin{array}{c|rrrrrr}1&1&0&0&5&\color{orangered}{ 1 }&-7\\& & 1& 1& 1& \color{orangered}{6} & \\ \hline &1&1&1&6&\color{orangered}{7}& \end{array} $$

Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 7 } = \color{blue}{ 7 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&0&0&5&1&-7\\& & 1& 1& 1& 6& \color{blue}{7} \\ \hline &1&1&1&6&\color{blue}{7}& \end{array} $$

Step 11 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 7 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrrr}1&1&0&0&5&1&\color{orangered}{ -7 }\\& & 1& 1& 1& 6& \color{orangered}{7} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{blue}{7}&\color{orangered}{0} \end{array} $$

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

Synthetic Division Calculator