Check if $ x-11 $ is a factor of $ x^{6}-90x^{5}+3345x^{4}-65812x^{3}+723536x^{2}-4214672x+10158192 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}11&1&-90&3345&-65812&723536&-4214672&10158192\\& & 11& -869& 27236& -424336& 3291200& \color{black}{-10158192} \\ \hline &\color{blue}{1}&\color{blue}{-79}&\color{blue}{2476}&\color{blue}{-38576}&\color{blue}{299200}&\color{blue}{-923472}&\color{orangered}{0} \end{array} $$Because the remainder equals zero, we conclude that the $ x-11 $ is a factor of the $ x^{6}-90x^{5}+3345x^{4}-65812x^{3}+723536x^{2}-4214672x+10158192 $.
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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{11}&1&-90&3345&-65812&723536&-4214672&10158192\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}11&\color{orangered}{ 1 }&-90&3345&-65812&723536&-4214672&10158192\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{11}&1&-90&3345&-65812&723536&-4214672&10158192\\& & \color{blue}{11} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -90 } + \color{orangered}{ 11 } = \color{orangered}{ -79 } $
$$ \begin{array}{c|rrrrrrr}11&1&\color{orangered}{ -90 }&3345&-65812&723536&-4214672&10158192\\& & \color{orangered}{11} & & & & & \\ \hline &1&\color{orangered}{-79}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ \left( -79 \right) } = \color{blue}{ -869 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{11}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 11& \color{blue}{-869} & & & & \\ \hline &1&\color{blue}{-79}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3345 } + \color{orangered}{ \left( -869 \right) } = \color{orangered}{ 2476 } $
$$ \begin{array}{c|rrrrrrr}11&1&-90&\color{orangered}{ 3345 }&-65812&723536&-4214672&10158192\\& & 11& \color{orangered}{-869} & & & & \\ \hline &1&-79&\color{orangered}{2476}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ 2476 } = \color{blue}{ 27236 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{11}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 11& -869& \color{blue}{27236} & & & \\ \hline &1&-79&\color{blue}{2476}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -65812 } + \color{orangered}{ 27236 } = \color{orangered}{ -38576 } $
$$ \begin{array}{c|rrrrrrr}11&1&-90&3345&\color{orangered}{ -65812 }&723536&-4214672&10158192\\& & 11& -869& \color{orangered}{27236} & & & \\ \hline &1&-79&2476&\color{orangered}{-38576}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ \left( -38576 \right) } = \color{blue}{ -424336 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{11}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 11& -869& 27236& \color{blue}{-424336} & & \\ \hline &1&-79&2476&\color{blue}{-38576}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 723536 } + \color{orangered}{ \left( -424336 \right) } = \color{orangered}{ 299200 } $
$$ \begin{array}{c|rrrrrrr}11&1&-90&3345&-65812&\color{orangered}{ 723536 }&-4214672&10158192\\& & 11& -869& 27236& \color{orangered}{-424336} & & \\ \hline &1&-79&2476&-38576&\color{orangered}{299200}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ 299200 } = \color{blue}{ 3291200 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{11}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 11& -869& 27236& -424336& \color{blue}{3291200} & \\ \hline &1&-79&2476&-38576&\color{blue}{299200}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -4214672 } + \color{orangered}{ 3291200 } = \color{orangered}{ -923472 } $
$$ \begin{array}{c|rrrrrrr}11&1&-90&3345&-65812&723536&\color{orangered}{ -4214672 }&10158192\\& & 11& -869& 27236& -424336& \color{orangered}{3291200} & \\ \hline &1&-79&2476&-38576&299200&\color{orangered}{-923472}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 11 } \cdot \color{blue}{ \left( -923472 \right) } = \color{blue}{ -10158192 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{11}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 11& -869& 27236& -424336& 3291200& \color{blue}{-10158192} \\ \hline &1&-79&2476&-38576&299200&\color{blue}{-923472}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 10158192 } + \color{orangered}{ \left( -10158192 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}11&1&-90&3345&-65812&723536&-4214672&\color{orangered}{ 10158192 }\\& & 11& -869& 27236& -424336& 3291200& \color{orangered}{-10158192} \\ \hline &\color{blue}{1}&\color{blue}{-79}&\color{blue}{2476}&\color{blue}{-38576}&\color{blue}{299200}&\color{blue}{-923472}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.