Find remainder when $ x^{7}+5x $ is divided by $ x-1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}1&1&0&0&0&0&0&5&0\\& & 1& 1& 1& 1& 1& 1& \color{black}{6} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{orangered}{6} \end{array} $$The remainder when $ x^{7}+5x $ is divided by $ x-1 $ is $ \, \color{red}{ 6 } $.
Explanation
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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}1&\color{orangered}{ 1 }&0&0&0&0&0&5&0\\& & & & & & & & \\ \hline &\color{orangered}{1}&&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & \color{blue}{1} & & & & & & \\ \hline &\color{blue}{1}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrrr}1&1&\color{orangered}{ 0 }&0&0&0&0&5&0\\& & \color{orangered}{1} & & & & & & \\ \hline &1&\color{orangered}{1}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & 1& \color{blue}{1} & & & & & \\ \hline &1&\color{blue}{1}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrrr}1&1&0&\color{orangered}{ 0 }&0&0&0&5&0\\& & 1& \color{orangered}{1} & & & & & \\ \hline &1&1&\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & 1& 1& \color{blue}{1} & & & & \\ \hline &1&1&\color{blue}{1}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrrr}1&1&0&0&\color{orangered}{ 0 }&0&0&5&0\\& & 1& 1& \color{orangered}{1} & & & & \\ \hline &1&1&1&\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & 1& 1& 1& \color{blue}{1} & & & \\ \hline &1&1&1&\color{blue}{1}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrrr}1&1&0&0&0&\color{orangered}{ 0 }&0&5&0\\& & 1& 1& 1& \color{orangered}{1} & & & \\ \hline &1&1&1&1&\color{orangered}{1}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & 1& 1& 1& 1& \color{blue}{1} & & \\ \hline &1&1&1&1&\color{blue}{1}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrrr}1&1&0&0&0&0&\color{orangered}{ 0 }&5&0\\& & 1& 1& 1& 1& \color{orangered}{1} & & \\ \hline &1&1&1&1&1&\color{orangered}{1}&& \end{array} $$Step 12 : 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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & 1& 1& 1& 1& 1& \color{blue}{1} & \\ \hline &1&1&1&1&1&\color{blue}{1}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 1 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrr}1&1&0&0&0&0&0&\color{orangered}{ 5 }&0\\& & 1& 1& 1& 1& 1& \color{orangered}{1} & \\ \hline &1&1&1&1&1&1&\color{orangered}{6}& \end{array} $$Step 14 : 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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{1}&1&0&0&0&0&0&5&0\\& & 1& 1& 1& 1& 1& 1& \color{blue}{6} \\ \hline &1&1&1&1&1&1&\color{blue}{6}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrr}1&1&0&0&0&0&0&5&\color{orangered}{ 0 }\\& & 1& 1& 1& 1& 1& 1& \color{orangered}{6} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{orangered}{6} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 6 }\right) $.