Check if $ x-22 $ is a factor of $ x^{6}-90x^{5}+3345x^{4}-65812x^{3}+723536x^{2}-4214672x+10158192 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}22&1&-90&3345&-65812&723536&-4214672&10158192\\& & 22& -1496& 40678& -552948& 3752936& \color{black}{-10158192} \\ \hline &\color{blue}{1}&\color{blue}{-68}&\color{blue}{1849}&\color{blue}{-25134}&\color{blue}{170588}&\color{blue}{-461736}&\color{orangered}{0} \end{array} $$Because the remainder equals zero, we conclude that the $ x-22 $ 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 -22 = 0 $ ( $ x = \color{blue}{ 22 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{22}&1&-90&3345&-65812&723536&-4214672&10158192\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}22&\color{orangered}{ 1 }&-90&3345&-65812&723536&-4214672&10158192\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 22 } \cdot \color{blue}{ 1 } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{22}&1&-90&3345&-65812&723536&-4214672&10158192\\& & \color{blue}{22} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -90 } + \color{orangered}{ 22 } = \color{orangered}{ -68 } $
$$ \begin{array}{c|rrrrrrr}22&1&\color{orangered}{ -90 }&3345&-65812&723536&-4214672&10158192\\& & \color{orangered}{22} & & & & & \\ \hline &1&\color{orangered}{-68}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 22 } \cdot \color{blue}{ \left( -68 \right) } = \color{blue}{ -1496 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{22}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 22& \color{blue}{-1496} & & & & \\ \hline &1&\color{blue}{-68}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3345 } + \color{orangered}{ \left( -1496 \right) } = \color{orangered}{ 1849 } $
$$ \begin{array}{c|rrrrrrr}22&1&-90&\color{orangered}{ 3345 }&-65812&723536&-4214672&10158192\\& & 22& \color{orangered}{-1496} & & & & \\ \hline &1&-68&\color{orangered}{1849}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 22 } \cdot \color{blue}{ 1849 } = \color{blue}{ 40678 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{22}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 22& -1496& \color{blue}{40678} & & & \\ \hline &1&-68&\color{blue}{1849}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -65812 } + \color{orangered}{ 40678 } = \color{orangered}{ -25134 } $
$$ \begin{array}{c|rrrrrrr}22&1&-90&3345&\color{orangered}{ -65812 }&723536&-4214672&10158192\\& & 22& -1496& \color{orangered}{40678} & & & \\ \hline &1&-68&1849&\color{orangered}{-25134}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 22 } \cdot \color{blue}{ \left( -25134 \right) } = \color{blue}{ -552948 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{22}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 22& -1496& 40678& \color{blue}{-552948} & & \\ \hline &1&-68&1849&\color{blue}{-25134}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 723536 } + \color{orangered}{ \left( -552948 \right) } = \color{orangered}{ 170588 } $
$$ \begin{array}{c|rrrrrrr}22&1&-90&3345&-65812&\color{orangered}{ 723536 }&-4214672&10158192\\& & 22& -1496& 40678& \color{orangered}{-552948} & & \\ \hline &1&-68&1849&-25134&\color{orangered}{170588}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 22 } \cdot \color{blue}{ 170588 } = \color{blue}{ 3752936 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{22}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 22& -1496& 40678& -552948& \color{blue}{3752936} & \\ \hline &1&-68&1849&-25134&\color{blue}{170588}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -4214672 } + \color{orangered}{ 3752936 } = \color{orangered}{ -461736 } $
$$ \begin{array}{c|rrrrrrr}22&1&-90&3345&-65812&723536&\color{orangered}{ -4214672 }&10158192\\& & 22& -1496& 40678& -552948& \color{orangered}{3752936} & \\ \hline &1&-68&1849&-25134&170588&\color{orangered}{-461736}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 22 } \cdot \color{blue}{ \left( -461736 \right) } = \color{blue}{ -10158192 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{22}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 22& -1496& 40678& -552948& 3752936& \color{blue}{-10158192} \\ \hline &1&-68&1849&-25134&170588&\color{blue}{-461736}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 10158192 } + \color{orangered}{ \left( -10158192 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}22&1&-90&3345&-65812&723536&-4214672&\color{orangered}{ 10158192 }\\& & 22& -1496& 40678& -552948& 3752936& \color{orangered}{-10158192} \\ \hline &\color{blue}{1}&\color{blue}{-68}&\color{blue}{1849}&\color{blue}{-25134}&\color{blue}{170588}&\color{blue}{-461736}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.