Check if $ x+3 $ is a factor of $ x^{4}-81 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}-3&1&0&0&0&-81\\& & -3& 9& -27& \color{black}{81} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{9}&\color{blue}{-27}&\color{orangered}{0} \end{array} $$

Because the remainder equals zero, we conclude that the $ x+3 $ is a factor of the $ x^{4}-81 $.

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 + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&0&0&-81\\& & & & & \\ \hline &&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&0&0&0&-81\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&0&0&-81\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ 0 }&0&0&-81\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-3}&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&0&0&-81\\& & -3& \color{blue}{9} & & \\ \hline &1&\color{blue}{-3}&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-3&1&0&\color{orangered}{ 0 }&0&-81\\& & -3& \color{orangered}{9} & & \\ \hline &1&-3&\color{orangered}{9}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 9 } = \color{blue}{ -27 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&0&0&-81\\& & -3& 9& \color{blue}{-27} & \\ \hline &1&-3&\color{blue}{9}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -27 } $

$$ \begin{array}{c|rrrrr}-3&1&0&0&\color{orangered}{ 0 }&-81\\& & -3& 9& \color{orangered}{-27} & \\ \hline &1&-3&9&\color{orangered}{-27}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ 81 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&0&0&-81\\& & -3& 9& -27& \color{blue}{81} \\ \hline &1&-3&9&\color{blue}{-27}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ -81 } + \color{orangered}{ 81 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}-3&1&0&0&0&\color{orangered}{ -81 }\\& & -3& 9& -27& \color{orangered}{81} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{9}&\color{blue}{-27}&\color{orangered}{0} \end{array} $$

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

Synthetic Division Calculator