Check if $ x+1 $ is a factor of $ x^{3}-4x+1 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-1&1&0&-4&1\\& & -1& 1& \color{black}{3} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-3}&\color{orangered}{4} \end{array} $$

Because the remainder $ \left( \color{red}{ 4 } \right) $ is not zero, we conclude that the $ x+1 $ is not a factor of $ x^{3}-4x+1$.

First we need to create a synthetic division table.

Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.

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

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

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

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

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

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -1 } $

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

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.

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

Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 1 } = \color{orangered}{ -3 } $

$$ \begin{array}{c|rrrr}-1&1&0&\color{orangered}{ -4 }&1\\& & -1& \color{orangered}{1} & \\ \hline &1&-1&\color{orangered}{-3}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 3 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&0&-4&1\\& & -1& 1& \color{blue}{3} \\ \hline &1&-1&\color{blue}{-3}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 3 } = \color{orangered}{ 4 } $

$$ \begin{array}{c|rrrr}-1&1&0&-4&\color{orangered}{ 1 }\\& & -1& 1& \color{orangered}{3} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-3}&\color{orangered}{4} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 4 }\right)$.

Synthetic Division Calculator