Check if $ x-18 $ is a factor of $ x^{6}-90x^{5}+3345x^{4}-65812x^{3}+723536x^{2}-4214672x+10158192 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}18&1&-90&3345&-65812&723536&-4214672&10158192\\& & 18& -1296& 36882& -520740& 3650328& \color{black}{-10158192} \\ \hline &\color{blue}{1}&\color{blue}{-72}&\color{blue}{2049}&\color{blue}{-28930}&\color{blue}{202796}&\color{blue}{-564344}&\color{orangered}{0} \end{array} $$Because the remainder equals zero, we conclude that the $ x-18 $ 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 -18 = 0 $ ( $ x = \color{blue}{ 18 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{18}&1&-90&3345&-65812&723536&-4214672&10158192\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}18&\color{orangered}{ 1 }&-90&3345&-65812&723536&-4214672&10158192\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 1 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{18}&1&-90&3345&-65812&723536&-4214672&10158192\\& & \color{blue}{18} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -90 } + \color{orangered}{ 18 } = \color{orangered}{ -72 } $
$$ \begin{array}{c|rrrrrrr}18&1&\color{orangered}{ -90 }&3345&-65812&723536&-4214672&10158192\\& & \color{orangered}{18} & & & & & \\ \hline &1&\color{orangered}{-72}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ \left( -72 \right) } = \color{blue}{ -1296 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{18}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 18& \color{blue}{-1296} & & & & \\ \hline &1&\color{blue}{-72}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3345 } + \color{orangered}{ \left( -1296 \right) } = \color{orangered}{ 2049 } $
$$ \begin{array}{c|rrrrrrr}18&1&-90&\color{orangered}{ 3345 }&-65812&723536&-4214672&10158192\\& & 18& \color{orangered}{-1296} & & & & \\ \hline &1&-72&\color{orangered}{2049}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 2049 } = \color{blue}{ 36882 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{18}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 18& -1296& \color{blue}{36882} & & & \\ \hline &1&-72&\color{blue}{2049}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -65812 } + \color{orangered}{ 36882 } = \color{orangered}{ -28930 } $
$$ \begin{array}{c|rrrrrrr}18&1&-90&3345&\color{orangered}{ -65812 }&723536&-4214672&10158192\\& & 18& -1296& \color{orangered}{36882} & & & \\ \hline &1&-72&2049&\color{orangered}{-28930}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ \left( -28930 \right) } = \color{blue}{ -520740 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{18}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 18& -1296& 36882& \color{blue}{-520740} & & \\ \hline &1&-72&2049&\color{blue}{-28930}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 723536 } + \color{orangered}{ \left( -520740 \right) } = \color{orangered}{ 202796 } $
$$ \begin{array}{c|rrrrrrr}18&1&-90&3345&-65812&\color{orangered}{ 723536 }&-4214672&10158192\\& & 18& -1296& 36882& \color{orangered}{-520740} & & \\ \hline &1&-72&2049&-28930&\color{orangered}{202796}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 202796 } = \color{blue}{ 3650328 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{18}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 18& -1296& 36882& -520740& \color{blue}{3650328} & \\ \hline &1&-72&2049&-28930&\color{blue}{202796}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -4214672 } + \color{orangered}{ 3650328 } = \color{orangered}{ -564344 } $
$$ \begin{array}{c|rrrrrrr}18&1&-90&3345&-65812&723536&\color{orangered}{ -4214672 }&10158192\\& & 18& -1296& 36882& -520740& \color{orangered}{3650328} & \\ \hline &1&-72&2049&-28930&202796&\color{orangered}{-564344}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ \left( -564344 \right) } = \color{blue}{ -10158192 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{18}&1&-90&3345&-65812&723536&-4214672&10158192\\& & 18& -1296& 36882& -520740& 3650328& \color{blue}{-10158192} \\ \hline &1&-72&2049&-28930&202796&\color{blue}{-564344}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 10158192 } + \color{orangered}{ \left( -10158192 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}18&1&-90&3345&-65812&723536&-4214672&\color{orangered}{ 10158192 }\\& & 18& -1296& 36882& -520740& 3650328& \color{orangered}{-10158192} \\ \hline &\color{blue}{1}&\color{blue}{-72}&\color{blue}{2049}&\color{blue}{-28930}&\color{blue}{202796}&\color{blue}{-564344}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.