Check if $ x-20 $ is a factor of $ x^{3}-2x^{2}-23x-60 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}20&1&-2&-23&-60\\& & 20& 360& \color{black}{6740} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{337}&\color{orangered}{6680} \end{array} $$Because the remainder $ \left( \color{red}{ 6680 } \right) $ is not zero, we conclude that the $ x-20 $ is not a factor of $ x^{3}-2x^{2}-23x-60$.
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 -20 = 0 $ ( $ x = \color{blue}{ 20 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-2&-23&-60\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}20&\color{orangered}{ 1 }&-2&-23&-60\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 1 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-2&-23&-60\\& & \color{blue}{20} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 20 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}20&1&\color{orangered}{ -2 }&-23&-60\\& & \color{orangered}{20} & & \\ \hline &1&\color{orangered}{18}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 18 } = \color{blue}{ 360 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-2&-23&-60\\& & 20& \color{blue}{360} & \\ \hline &1&\color{blue}{18}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 360 } = \color{orangered}{ 337 } $
$$ \begin{array}{c|rrrr}20&1&-2&\color{orangered}{ -23 }&-60\\& & 20& \color{orangered}{360} & \\ \hline &1&18&\color{orangered}{337}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 337 } = \color{blue}{ 6740 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&1&-2&-23&-60\\& & 20& 360& \color{blue}{6740} \\ \hline &1&18&\color{blue}{337}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -60 } + \color{orangered}{ 6740 } = \color{orangered}{ 6680 } $
$$ \begin{array}{c|rrrr}20&1&-2&-23&\color{orangered}{ -60 }\\& & 20& 360& \color{orangered}{6740} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{337}&\color{orangered}{6680} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 6680 }\right)$.