Find remainder when $ 9x^{4}-3x^{3}+72x-17 $ is divided by $ 9x-3 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&9&-3&0&72&-17\\& & 27& 72& 216& \color{black}{864} \\ \hline &\color{blue}{9}&\color{blue}{24}&\color{blue}{72}&\color{blue}{288}&\color{orangered}{847} \end{array} $$The remainder when $ 9x^{4}-3x^{3}+72x-17 $ is divided by $ x-3 $ is $ \, \color{red}{ 847 } $.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&9&-3&0&72&-17\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 9 }&-3&0&72&-17\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 9 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&9&-3&0&72&-17\\& & \color{blue}{27} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 27 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrr}3&9&\color{orangered}{ -3 }&0&72&-17\\& & \color{orangered}{27} & & & \\ \hline &9&\color{orangered}{24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 24 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&9&-3&0&72&-17\\& & 27& \color{blue}{72} & & \\ \hline &9&\color{blue}{24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 72 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrrr}3&9&-3&\color{orangered}{ 0 }&72&-17\\& & 27& \color{orangered}{72} & & \\ \hline &9&24&\color{orangered}{72}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 72 } = \color{blue}{ 216 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&9&-3&0&72&-17\\& & 27& 72& \color{blue}{216} & \\ \hline &9&24&\color{blue}{72}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 72 } + \color{orangered}{ 216 } = \color{orangered}{ 288 } $
$$ \begin{array}{c|rrrrr}3&9&-3&0&\color{orangered}{ 72 }&-17\\& & 27& 72& \color{orangered}{216} & \\ \hline &9&24&72&\color{orangered}{288}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 288 } = \color{blue}{ 864 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&9&-3&0&72&-17\\& & 27& 72& 216& \color{blue}{864} \\ \hline &9&24&72&\color{blue}{288}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 864 } = \color{orangered}{ 847 } $
$$ \begin{array}{c|rrrrr}3&9&-3&0&72&\color{orangered}{ -17 }\\& & 27& 72& 216& \color{orangered}{864} \\ \hline &\color{blue}{9}&\color{blue}{24}&\color{blue}{72}&\color{blue}{288}&\color{orangered}{847} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 847 }\right) $.