Find remainder when $ x^{9}+2x^{8}+3x^{7}+9x $ is divided by $ x-1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& 6& 6& 6& 6& 6& 6& \color{black}{15} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{15}&\color{orangered}{15} \end{array} $$The remainder when $ x^{9}+2x^{8}+3x^{7}+9x $ is divided by $ x-1 $ is $ \, \color{red}{ 15 } $.
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.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & & & & & & & & & \\ \hline &&&&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrrrr}1&\color{orangered}{ 1 }&2&3&0&0&0&0&0&9&0\\& & & & & & & & & & \\ \hline &\color{orangered}{1}&&&&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & \color{blue}{1} & & & & & & & & \\ \hline &\color{blue}{1}&&&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 1 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&\color{orangered}{ 2 }&3&0&0&0&0&0&9&0\\& & \color{orangered}{1} & & & & & & & & \\ \hline &1&\color{orangered}{3}&&&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 3 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& \color{blue}{3} & & & & & & & \\ \hline &1&\color{blue}{3}&&&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 3 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&\color{orangered}{ 3 }&0&0&0&0&0&9&0\\& & 1& \color{orangered}{3} & & & & & & & \\ \hline &1&3&\color{orangered}{6}&&&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& \color{blue}{6} & & & & & & \\ \hline &1&3&\color{blue}{6}&&&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&\color{orangered}{ 0 }&0&0&0&0&9&0\\& & 1& 3& \color{orangered}{6} & & & & & & \\ \hline &1&3&6&\color{orangered}{6}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& 6& \color{blue}{6} & & & & & \\ \hline &1&3&6&\color{blue}{6}&&&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&0&\color{orangered}{ 0 }&0&0&0&9&0\\& & 1& 3& 6& \color{orangered}{6} & & & & & \\ \hline &1&3&6&6&\color{orangered}{6}&&&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& 6& 6& \color{blue}{6} & & & & \\ \hline &1&3&6&6&\color{blue}{6}&&&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&0&0&\color{orangered}{ 0 }&0&0&9&0\\& & 1& 3& 6& 6& \color{orangered}{6} & & & & \\ \hline &1&3&6&6&6&\color{orangered}{6}&&&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& 6& 6& 6& \color{blue}{6} & & & \\ \hline &1&3&6&6&6&\color{blue}{6}&&&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&0&0&0&\color{orangered}{ 0 }&0&9&0\\& & 1& 3& 6& 6& 6& \color{orangered}{6} & & & \\ \hline &1&3&6&6&6&6&\color{orangered}{6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& 6& 6& 6& 6& \color{blue}{6} & & \\ \hline &1&3&6&6&6&6&\color{blue}{6}&&& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&0&0&0&0&\color{orangered}{ 0 }&9&0\\& & 1& 3& 6& 6& 6& 6& \color{orangered}{6} & & \\ \hline &1&3&6&6&6&6&6&\color{orangered}{6}&& \end{array} $$Step 16 : 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 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& 6& 6& 6& 6& 6& \color{blue}{6} & \\ \hline &1&3&6&6&6&6&6&\color{blue}{6}&& \end{array} $$Step 17 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 6 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&0&0&0&0&0&\color{orangered}{ 9 }&0\\& & 1& 3& 6& 6& 6& 6& 6& \color{orangered}{6} & \\ \hline &1&3&6&6&6&6&6&6&\color{orangered}{15}& \end{array} $$Step 18 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 15 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrrrrrrr}\color{blue}{1}&1&2&3&0&0&0&0&0&9&0\\& & 1& 3& 6& 6& 6& 6& 6& 6& \color{blue}{15} \\ \hline &1&3&6&6&6&6&6&6&\color{blue}{15}& \end{array} $$Step 19 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 15 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrrrrrrr}1&1&2&3&0&0&0&0&0&9&\color{orangered}{ 0 }\\& & 1& 3& 6& 6& 6& 6& 6& 6& \color{orangered}{15} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{6}&\color{blue}{15}&\color{orangered}{15} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 15 }\right) $.