Find remainder when $ x^{4}-4x^{3}-x+5 $ is divided by $ 3 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}0&1&-4&0&-1&5\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{5} \end{array} $$

The remainder when $ x^{4}-4x^{3}-x+5 $ is divided by $ x $ is $ \, \color{red}{ 5 } $.

We can find remainder using synthetic division method.

Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.

$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-4&0&-1&5\\& & & & & \\ \hline &&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 1 }&-4&0&-1&5\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-4&0&-1&5\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 0 } = \color{orangered}{ -4 } $

$$ \begin{array}{c|rrrrr}0&1&\color{orangered}{ -4 }&0&-1&5\\& & \color{orangered}{0} & & & \\ \hline &1&\color{orangered}{-4}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-4&0&-1&5\\& & 0& \color{blue}{0} & & \\ \hline &1&\color{blue}{-4}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}0&1&-4&\color{orangered}{ 0 }&-1&5\\& & 0& \color{orangered}{0} & & \\ \hline &1&-4&\color{orangered}{0}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-4&0&-1&5\\& & 0& 0& \color{blue}{0} & \\ \hline &1&-4&\color{blue}{0}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 0 } = \color{orangered}{ -1 } $

$$ \begin{array}{c|rrrrr}0&1&-4&0&\color{orangered}{ -1 }&5\\& & 0& 0& \color{orangered}{0} & \\ \hline &1&-4&0&\color{orangered}{-1}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&-4&0&-1&5\\& & 0& 0& 0& \color{blue}{0} \\ \hline &1&-4&0&\color{blue}{-1}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 0 } = \color{orangered}{ 5 } $

$$ \begin{array}{c|rrrrr}0&1&-4&0&-1&\color{orangered}{ 5 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{5} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 5 }\right) $.

Synthetic Division Calculator