Find remainder when $ x^{4}-4x^{2}+2x+10 $ is divided by $ x-2 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}2&1&0&-4&2&10\\& & 2& 4& 0& \color{black}{4} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{0}&\color{blue}{2}&\color{orangered}{14} \end{array} $$

The remainder when $ x^{4}-4x^{2}+2x+10 $ is divided by $ x-2 $ is $ \, \color{red}{ 14 } $.

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 -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.

$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&0&-4&2&10\\& & & & & \\ \hline &&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 1 }&0&-4&2&10\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&0&-4&2&10\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $

$$ \begin{array}{c|rrrrr}2&1&\color{orangered}{ 0 }&-4&2&10\\& & \color{orangered}{2} & & & \\ \hline &1&\color{orangered}{2}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&0&-4&2&10\\& & 2& \color{blue}{4} & & \\ \hline &1&\color{blue}{2}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}2&1&0&\color{orangered}{ -4 }&2&10\\& & 2& \color{orangered}{4} & & \\ \hline &1&2&\color{orangered}{0}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&0&-4&2&10\\& & 2& 4& \color{blue}{0} & \\ \hline &1&2&\color{blue}{0}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $

$$ \begin{array}{c|rrrrr}2&1&0&-4&\color{orangered}{ 2 }&10\\& & 2& 4& \color{orangered}{0} & \\ \hline &1&2&0&\color{orangered}{2}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&0&-4&2&10\\& & 2& 4& 0& \color{blue}{4} \\ \hline &1&2&0&\color{blue}{2}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 4 } = \color{orangered}{ 14 } $

$$ \begin{array}{c|rrrrr}2&1&0&-4&2&\color{orangered}{ 10 }\\& & 2& 4& 0& \color{orangered}{4} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{0}&\color{blue}{2}&\color{orangered}{14} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 14 }\right) $.

Synthetic Division Calculator