Check if $ x-1 $ is a factor of $ x^{3}-2x^{2}-23x-60 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&1&-2&-23&-60\\& & 1& -1& \color{black}{-24} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-24}&\color{orangered}{-84} \end{array} $$Because the remainder $ \left( \color{red}{ -84 } \right) $ is not zero, we conclude that the $ x-1 $ 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 -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-2&-23&-60\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 1 }&-2&-23&-60\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-2&-23&-60\\& & \color{blue}{1} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 1 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}1&1&\color{orangered}{ -2 }&-23&-60\\& & \color{orangered}{1} & & \\ \hline &1&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-2&-23&-60\\& & 1& \color{blue}{-1} & \\ \hline &1&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}1&1&-2&\color{orangered}{ -23 }&-60\\& & 1& \color{orangered}{-1} & \\ \hline &1&-1&\color{orangered}{-24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&-2&-23&-60\\& & 1& -1& \color{blue}{-24} \\ \hline &1&-1&\color{blue}{-24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -60 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -84 } $
$$ \begin{array}{c|rrrr}1&1&-2&-23&\color{orangered}{ -60 }\\& & 1& -1& \color{orangered}{-24} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-24}&\color{orangered}{-84} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -84 }\right)$.