Check if $ x-3 $ is a factor of $ -x^{3}+8x+20 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}3&-1&0&8&20\\& & -3& -9& \color{black}{-3} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{blue}{-1}&\color{orangered}{17} \end{array} $$

Because the remainder $ \left( \color{red}{ 17 } \right) $ is not zero, we conclude that the $ x-3 $ is not a factor of $ -x^{3}+8x+20$.

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.

$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&0&8&20\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}3&\color{orangered}{ -1 }&0&8&20\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&0&8&20\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}3&-1&\color{orangered}{ 0 }&8&20\\& & \color{orangered}{-3} & & \\ \hline &-1&\color{orangered}{-3}&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&0&8&20\\& & -3& \color{blue}{-9} & \\ \hline &-1&\color{blue}{-3}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -1 } $

$$ \begin{array}{c|rrrr}3&-1&0&\color{orangered}{ 8 }&20\\& & -3& \color{orangered}{-9} & \\ \hline &-1&-3&\color{orangered}{-1}& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&0&8&20\\& & -3& -9& \color{blue}{-3} \\ \hline &-1&-3&\color{blue}{-1}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 17 } $

$$ \begin{array}{c|rrrr}3&-1&0&8&\color{orangered}{ 20 }\\& & -3& -9& \color{orangered}{-3} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{blue}{-1}&\color{orangered}{17} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 17 }\right)$.

Synthetic Division Calculator