Check if $ x+60 $ is a factor of $ x^{3}-2x^{2}-23x-60 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-60&1&-2&-23&-60\\& & -60& 3720& \color{black}{-221820} \\ \hline &\color{blue}{1}&\color{blue}{-62}&\color{blue}{3697}&\color{orangered}{-221880} \end{array} $$Because the remainder $ \left( \color{red}{ -221880 } \right) $ is not zero, we conclude that the $ x+60 $ 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 + 60 = 0 $ ( $ x = \color{blue}{ -60 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-60}&1&-2&-23&-60\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-60&\color{orangered}{ 1 }&-2&-23&-60\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -60 } \cdot \color{blue}{ 1 } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-60}&1&-2&-23&-60\\& & \color{blue}{-60} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -62 } $
$$ \begin{array}{c|rrrr}-60&1&\color{orangered}{ -2 }&-23&-60\\& & \color{orangered}{-60} & & \\ \hline &1&\color{orangered}{-62}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -60 } \cdot \color{blue}{ \left( -62 \right) } = \color{blue}{ 3720 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-60}&1&-2&-23&-60\\& & -60& \color{blue}{3720} & \\ \hline &1&\color{blue}{-62}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 3720 } = \color{orangered}{ 3697 } $
$$ \begin{array}{c|rrrr}-60&1&-2&\color{orangered}{ -23 }&-60\\& & -60& \color{orangered}{3720} & \\ \hline &1&-62&\color{orangered}{3697}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -60 } \cdot \color{blue}{ 3697 } = \color{blue}{ -221820 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-60}&1&-2&-23&-60\\& & -60& 3720& \color{blue}{-221820} \\ \hline &1&-62&\color{blue}{3697}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -60 } + \color{orangered}{ \left( -221820 \right) } = \color{orangered}{ -221880 } $
$$ \begin{array}{c|rrrr}-60&1&-2&-23&\color{orangered}{ -60 }\\& & -60& 3720& \color{orangered}{-221820} \\ \hline &\color{blue}{1}&\color{blue}{-62}&\color{blue}{3697}&\color{orangered}{-221880} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -221880 }\right)$.