Check if $ x-11 $ is a factor of $ x^{4}-19x^{3}+104x^{2}+140x-54 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}11&1&-19&104&140&-54\\& & 11& -88& 176& \color{black}{3476} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{16}&\color{blue}{316}&\color{orangered}{3422} \end{array} $$Because the remainder $ \left( \color{red}{ 3422 } \right) $ is not zero, we conclude that the $ x-11 $ is not a factor of $ x^{4}-19x^{3}+104x^{2}+140x-54$.
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 -11 = 0 $ ( $ x = \color{blue}{ 11 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{11}&1&-19&104&140&-54\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}11&\color{orangered}{ 1 }&-19&104&140&-54\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ 1 } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{11}&1&-19&104&140&-54\\& & \color{blue}{11} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 11 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}11&1&\color{orangered}{ -19 }&104&140&-54\\& & \color{orangered}{11} & & & \\ \hline &1&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -88 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{11}&1&-19&104&140&-54\\& & 11& \color{blue}{-88} & & \\ \hline &1&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 104 } + \color{orangered}{ \left( -88 \right) } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}11&1&-19&\color{orangered}{ 104 }&140&-54\\& & 11& \color{orangered}{-88} & & \\ \hline &1&-8&\color{orangered}{16}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ 16 } = \color{blue}{ 176 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{11}&1&-19&104&140&-54\\& & 11& -88& \color{blue}{176} & \\ \hline &1&-8&\color{blue}{16}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 140 } + \color{orangered}{ 176 } = \color{orangered}{ 316 } $
$$ \begin{array}{c|rrrrr}11&1&-19&104&\color{orangered}{ 140 }&-54\\& & 11& -88& \color{orangered}{176} & \\ \hline &1&-8&16&\color{orangered}{316}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ 316 } = \color{blue}{ 3476 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{11}&1&-19&104&140&-54\\& & 11& -88& 176& \color{blue}{3476} \\ \hline &1&-8&16&\color{blue}{316}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 3476 } = \color{orangered}{ 3422 } $
$$ \begin{array}{c|rrrrr}11&1&-19&104&140&\color{orangered}{ -54 }\\& & 11& -88& 176& \color{orangered}{3476} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{16}&\color{blue}{316}&\color{orangered}{3422} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 3422 }\right)$.