Check if $ x+20 $ is a factor of $ x^{3}-2x^{2}-23x-60 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-20&1&-2&-23&-60\\& & -20& 440& \color{black}{-8340} \\ \hline &\color{blue}{1}&\color{blue}{-22}&\color{blue}{417}&\color{orangered}{-8400} \end{array} $$Because the remainder $ \left( \color{red}{ -8400 } \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}{ \left( -20 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrrr}-20&1&\color{orangered}{ -2 }&-23&-60\\& & \color{orangered}{-20} & & \\ \hline &1&\color{orangered}{-22}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -20 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ 440 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-20}&1&-2&-23&-60\\& & -20& \color{blue}{440} & \\ \hline &1&\color{blue}{-22}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 440 } = \color{orangered}{ 417 } $
$$ \begin{array}{c|rrrr}-20&1&-2&\color{orangered}{ -23 }&-60\\& & -20& \color{orangered}{440} & \\ \hline &1&-22&\color{orangered}{417}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -20 } \cdot \color{blue}{ 417 } = \color{blue}{ -8340 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-20}&1&-2&-23&-60\\& & -20& 440& \color{blue}{-8340} \\ \hline &1&-22&\color{blue}{417}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -60 } + \color{orangered}{ \left( -8340 \right) } = \color{orangered}{ -8400 } $
$$ \begin{array}{c|rrrr}-20&1&-2&-23&\color{orangered}{ -60 }\\& & -20& 440& \color{orangered}{-8340} \\ \hline &\color{blue}{1}&\color{blue}{-22}&\color{blue}{417}&\color{orangered}{-8400} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -8400 }\right)$.