Find remainder when $ x^{2}-4x+1 $ is divided by $ x+2 $.

The synthetic division table is:

$$ \begin{array}{c|rrr}-2&1&-4&1\\& & -2& \color{black}{12} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{orangered}{13} \end{array} $$

The remainder when $ x^{2}-4x+1 $ is divided by $ x+2 $ is $ \, \color{red}{ 13 } $.

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.

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

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

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

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 } $.

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

Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -6 } $

$$ \begin{array}{c|rrr}-2&1&\color{orangered}{ -4 }&1\\& & \color{orangered}{-2} & \\ \hline &1&\color{orangered}{-6}& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 12 } $.

$$ \begin{array}{c|rrr}\color{blue}{-2}&1&-4&1\\& & -2& \color{blue}{12} \\ \hline &1&\color{blue}{-6}& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 12 } = \color{orangered}{ 13 } $

$$ \begin{array}{c|rrr}-2&1&-4&\color{orangered}{ 1 }\\& & -2& \color{orangered}{12} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{orangered}{13} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 13 }\right) $.

Synthetic Division Calculator