Find remainder when $ 27x^{3}-3x+5 $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&27&0&-3&5\\& & 81& 243& \color{black}{720} \\ \hline &\color{blue}{27}&\color{blue}{81}&\color{blue}{240}&\color{orangered}{725} \end{array} $$The remainder when $ 27x^{3}-3x+5 $ is divided by $ x-3 $ is $ \, \color{red}{ 725 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&27&0&-3&5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 27 }&0&-3&5\\& & & & \\ \hline &\color{orangered}{27}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 27 } = \color{blue}{ 81 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&27&0&-3&5\\& & \color{blue}{81} & & \\ \hline &\color{blue}{27}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 81 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrr}3&27&\color{orangered}{ 0 }&-3&5\\& & \color{orangered}{81} & & \\ \hline &27&\color{orangered}{81}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 81 } = \color{blue}{ 243 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&27&0&-3&5\\& & 81& \color{blue}{243} & \\ \hline &27&\color{blue}{81}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 243 } = \color{orangered}{ 240 } $
$$ \begin{array}{c|rrrr}3&27&0&\color{orangered}{ -3 }&5\\& & 81& \color{orangered}{243} & \\ \hline &27&81&\color{orangered}{240}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 240 } = \color{blue}{ 720 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&27&0&-3&5\\& & 81& 243& \color{blue}{720} \\ \hline &27&81&\color{blue}{240}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 720 } = \color{orangered}{ 725 } $
$$ \begin{array}{c|rrrr}3&27&0&-3&\color{orangered}{ 5 }\\& & 81& 243& \color{orangered}{720} \\ \hline &\color{blue}{27}&\color{blue}{81}&\color{blue}{240}&\color{orangered}{725} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 725 }\right) $.