Check if $ 5x-1 $ is a factor of $ 5x^{3}-9x^{2}+28x+6 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}1&5&-9&28&6\\& & 5& -4& \color{black}{24} \\ \hline &\color{blue}{5}&\color{blue}{-4}&\color{blue}{24}&\color{orangered}{30} \end{array} $$

Because the remainder $ \left( \color{red}{ 30 } \right) $ is not zero, we conclude that the $ x-1 $ is not a factor of $ 5x^{3}-9x^{2}+28x+6$.

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.

$$ \begin{array}{c|rrrr}\color{blue}{1}&5&-9&28&6\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}1&\color{orangered}{ 5 }&-9&28&6\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 5 } = \color{blue}{ 5 } $.

$$ \begin{array}{c|rrrr}\color{blue}{1}&5&-9&28&6\\& & \color{blue}{5} & & \\ \hline &\color{blue}{5}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 5 } = \color{orangered}{ -4 } $

$$ \begin{array}{c|rrrr}1&5&\color{orangered}{ -9 }&28&6\\& & \color{orangered}{5} & & \\ \hline &5&\color{orangered}{-4}&& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{1}&5&-9&28&6\\& & 5& \color{blue}{-4} & \\ \hline &5&\color{blue}{-4}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 28 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 24 } $

$$ \begin{array}{c|rrrr}1&5&-9&\color{orangered}{ 28 }&6\\& & 5& \color{orangered}{-4} & \\ \hline &5&-4&\color{orangered}{24}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 24 } = \color{blue}{ 24 } $.

$$ \begin{array}{c|rrrr}\color{blue}{1}&5&-9&28&6\\& & 5& -4& \color{blue}{24} \\ \hline &5&-4&\color{blue}{24}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 24 } = \color{orangered}{ 30 } $

$$ \begin{array}{c|rrrr}1&5&-9&28&\color{orangered}{ 6 }\\& & 5& -4& \color{orangered}{24} \\ \hline &\color{blue}{5}&\color{blue}{-4}&\color{blue}{24}&\color{orangered}{30} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 30 }\right)$.

Synthetic Division Calculator