Check if $ x-7 $ is a factor of $ -x^{3}+7x^{2}-6x-72 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&-1&7&-6&-72\\& & -7& 0& \color{black}{-42} \\ \hline &\color{blue}{-1}&\color{blue}{0}&\color{blue}{-6}&\color{orangered}{-114} \end{array} $$Because the remainder $ \left( \color{red}{ -114 } \right) $ is not zero, we conclude that the $ x-7 $ is not a factor of $ -x^{3}+7x^{2}-6x-72$.
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 -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-1&7&-6&-72\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ -1 }&7&-6&-72\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-1&7&-6&-72\\& & \color{blue}{-7} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}7&-1&\color{orangered}{ 7 }&-6&-72\\& & \color{orangered}{-7} & & \\ \hline &-1&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-1&7&-6&-72\\& & -7& \color{blue}{0} & \\ \hline &-1&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 0 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}7&-1&7&\color{orangered}{ -6 }&-72\\& & -7& \color{orangered}{0} & \\ \hline &-1&0&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-1&7&-6&-72\\& & -7& 0& \color{blue}{-42} \\ \hline &-1&0&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -72 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -114 } $
$$ \begin{array}{c|rrrr}7&-1&7&-6&\color{orangered}{ -72 }\\& & -7& 0& \color{orangered}{-42} \\ \hline &\color{blue}{-1}&\color{blue}{0}&\color{blue}{-6}&\color{orangered}{-114} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -114 }\right)$.