Check if $ x-5 $ is a factor of $ x^{3}-125 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}5&1&0&0&-125\\& & 5& 25& \color{black}{125} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{25}&\color{orangered}{0} \end{array} $$

Because the remainder equals zero, we conclude that the $ x-5 $ is a factor of the $ x^{3}-125 $.

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 -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&0&-125\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&0&0&-125\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&0&-125\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $

$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ 0 }&0&-125\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{5}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5 } = \color{blue}{ 25 } $.

$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&0&-125\\& & 5& \color{blue}{25} & \\ \hline &1&\color{blue}{5}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 25 } = \color{orangered}{ 25 } $

$$ \begin{array}{c|rrrr}5&1&0&\color{orangered}{ 0 }&-125\\& & 5& \color{orangered}{25} & \\ \hline &1&5&\color{orangered}{25}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 25 } = \color{blue}{ 125 } $.

$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&0&-125\\& & 5& 25& \color{blue}{125} \\ \hline &1&5&\color{blue}{25}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -125 } + \color{orangered}{ 125 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrr}5&1&0&0&\color{orangered}{ -125 }\\& & 5& 25& \color{orangered}{125} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{25}&\color{orangered}{0} \end{array} $$

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

Synthetic Division Calculator