Check if $ x-5 $ is a factor of $ 4x^{3}-12x^{2}-30x+25 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&4&-12&-30&25\\& & 20& 40& \color{black}{50} \\ \hline &\color{blue}{4}&\color{blue}{8}&\color{blue}{10}&\color{orangered}{75} \end{array} $$Because the remainder $ \left( \color{red}{ 75 } \right) $ is not zero, we conclude that the $ x-5 $ is not a factor of $ 4x^{3}-12x^{2}-30x+25$.
Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-12&-30&25\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 4 }&-12&-30&25\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 4 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-12&-30&25\\& & \color{blue}{20} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 20 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}5&4&\color{orangered}{ -12 }&-30&25\\& & \color{orangered}{20} & & \\ \hline &4&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 8 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-12&-30&25\\& & 20& \color{blue}{40} & \\ \hline &4&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 40 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}5&4&-12&\color{orangered}{ -30 }&25\\& & 20& \color{orangered}{40} & \\ \hline &4&8&\color{orangered}{10}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 10 } = \color{blue}{ 50 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-12&-30&25\\& & 20& 40& \color{blue}{50} \\ \hline &4&8&\color{blue}{10}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ 50 } = \color{orangered}{ 75 } $
$$ \begin{array}{c|rrrr}5&4&-12&-30&\color{orangered}{ 25 }\\& & 20& 40& \color{orangered}{50} \\ \hline &\color{blue}{4}&\color{blue}{8}&\color{blue}{10}&\color{orangered}{75} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 75 }\right)$.