Find remainder when $ x^{6}+9x+3 $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}3&1&0&0&0&0&9&3\\& & 3& 9& 27& 81& 243& \color{black}{756} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{9}&\color{blue}{27}&\color{blue}{81}&\color{blue}{252}&\color{orangered}{759} \end{array} $$The remainder when $ x^{6}+9x+3 $ is divided by $ x-3 $ is $ \, \color{red}{ 759 } $.
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 -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&0&0&0&9&3\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}3&\color{orangered}{ 1 }&0&0&0&0&9&3\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&0&0&0&9&3\\& & \color{blue}{3} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 3 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrr}3&1&\color{orangered}{ 0 }&0&0&0&9&3\\& & \color{orangered}{3} & & & & & \\ \hline &1&\color{orangered}{3}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&0&0&0&9&3\\& & 3& \color{blue}{9} & & & & \\ \hline &1&\color{blue}{3}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 9 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&\color{orangered}{ 0 }&0&0&9&3\\& & 3& \color{orangered}{9} & & & & \\ \hline &1&3&\color{orangered}{9}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 9 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&0&0&0&9&3\\& & 3& 9& \color{blue}{27} & & & \\ \hline &1&3&\color{blue}{9}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 27 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&0&\color{orangered}{ 0 }&0&9&3\\& & 3& 9& \color{orangered}{27} & & & \\ \hline &1&3&9&\color{orangered}{27}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 27 } = \color{blue}{ 81 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&0&0&0&9&3\\& & 3& 9& 27& \color{blue}{81} & & \\ \hline &1&3&9&\color{blue}{27}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 81 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&0&0&\color{orangered}{ 0 }&9&3\\& & 3& 9& 27& \color{orangered}{81} & & \\ \hline &1&3&9&27&\color{orangered}{81}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 81 } = \color{blue}{ 243 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&0&0&0&9&3\\& & 3& 9& 27& 81& \color{blue}{243} & \\ \hline &1&3&9&27&\color{blue}{81}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 243 } = \color{orangered}{ 252 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&0&0&0&\color{orangered}{ 9 }&3\\& & 3& 9& 27& 81& \color{orangered}{243} & \\ \hline &1&3&9&27&81&\color{orangered}{252}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 252 } = \color{blue}{ 756 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&0&0&0&9&3\\& & 3& 9& 27& 81& 243& \color{blue}{756} \\ \hline &1&3&9&27&81&\color{blue}{252}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 756 } = \color{orangered}{ 759 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&0&0&0&9&\color{orangered}{ 3 }\\& & 3& 9& 27& 81& 243& \color{orangered}{756} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{9}&\color{blue}{27}&\color{blue}{81}&\color{blue}{252}&\color{orangered}{759} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 759 }\right) $.