Find remainder when $ 3x^{4}-25x^{2}-8x+6 $ is divided by $ x-3 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}3&3&0&-25&-8&6\\& & 9& 27& 6& \color{black}{-6} \\ \hline &\color{blue}{3}&\color{blue}{9}&\color{blue}{2}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$

The remainder when $ 3x^{4}-25x^{2}-8x+6 $ is divided by $ x-3 $ is $ \, \color{red}{ 0 } $.

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.

$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&0&-25&-8&6\\& & & & & \\ \hline &&&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 3 }&0&-25&-8&6\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&0&-25&-8&6\\& & \color{blue}{9} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 9 } = \color{orangered}{ 9 } $

$$ \begin{array}{c|rrrrr}3&3&\color{orangered}{ 0 }&-25&-8&6\\& & \color{orangered}{9} & & & \\ \hline &3&\color{orangered}{9}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 9 } = \color{blue}{ 27 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&0&-25&-8&6\\& & 9& \color{blue}{27} & & \\ \hline &3&\color{blue}{9}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 27 } = \color{orangered}{ 2 } $

$$ \begin{array}{c|rrrrr}3&3&0&\color{orangered}{ -25 }&-8&6\\& & 9& \color{orangered}{27} & & \\ \hline &3&9&\color{orangered}{2}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&0&-25&-8&6\\& & 9& 27& \color{blue}{6} & \\ \hline &3&9&\color{blue}{2}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 6 } = \color{orangered}{ -2 } $

$$ \begin{array}{c|rrrrr}3&3&0&-25&\color{orangered}{ -8 }&6\\& & 9& 27& \color{orangered}{6} & \\ \hline &3&9&2&\color{orangered}{-2}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -6 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&0&-25&-8&6\\& & 9& 27& 6& \color{blue}{-6} \\ \hline &3&9&2&\color{blue}{-2}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}3&3&0&-25&-8&\color{orangered}{ 6 }\\& & 9& 27& 6& \color{orangered}{-6} \\ \hline &\color{blue}{3}&\color{blue}{9}&\color{blue}{2}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.

Synthetic Division Calculator