Divide using synthetic division $$ ( 9x^{6}-7x^{4}+7x^{2}+6 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&9&0&-7&0&7&0&6\\& & 18& 36& 58& 116& 246& \color{black}{492} \\ \hline &\color{blue}{9}&\color{blue}{18}&\color{blue}{29}&\color{blue}{58}&\color{blue}{123}&\color{blue}{246}&\color{orangered}{498} \end{array} $$The solution is:
$$ \frac{ 9x^{6}-7x^{4}+7x^{2}+6 }{ x-2 } = \color{blue}{9x^{5}+18x^{4}+29x^{3}+58x^{2}+123x+246} ~+~ \frac{ \color{red}{ 498 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-7&0&7&0&6\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 9 }&0&-7&0&7&0&6\\& & & & & & & \\ \hline &\color{orangered}{9}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-7&0&7&0&6\\& & \color{blue}{18} & & & & & \\ \hline &\color{blue}{9}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrrr}2&9&\color{orangered}{ 0 }&-7&0&7&0&6\\& & \color{orangered}{18} & & & & & \\ \hline &9&\color{orangered}{18}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 18 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-7&0&7&0&6\\& & 18& \color{blue}{36} & & & & \\ \hline &9&\color{blue}{18}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 36 } = \color{orangered}{ 29 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&\color{orangered}{ -7 }&0&7&0&6\\& & 18& \color{orangered}{36} & & & & \\ \hline &9&18&\color{orangered}{29}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 29 } = \color{blue}{ 58 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-7&0&7&0&6\\& & 18& 36& \color{blue}{58} & & & \\ \hline &9&18&\color{blue}{29}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 58 } = \color{orangered}{ 58 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-7&\color{orangered}{ 0 }&7&0&6\\& & 18& 36& \color{orangered}{58} & & & \\ \hline &9&18&29&\color{orangered}{58}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 58 } = \color{blue}{ 116 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-7&0&7&0&6\\& & 18& 36& 58& \color{blue}{116} & & \\ \hline &9&18&29&\color{blue}{58}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 116 } = \color{orangered}{ 123 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-7&0&\color{orangered}{ 7 }&0&6\\& & 18& 36& 58& \color{orangered}{116} & & \\ \hline &9&18&29&58&\color{orangered}{123}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 123 } = \color{blue}{ 246 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-7&0&7&0&6\\& & 18& 36& 58& 116& \color{blue}{246} & \\ \hline &9&18&29&58&\color{blue}{123}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 246 } = \color{orangered}{ 246 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-7&0&7&\color{orangered}{ 0 }&6\\& & 18& 36& 58& 116& \color{orangered}{246} & \\ \hline &9&18&29&58&123&\color{orangered}{246}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 246 } = \color{blue}{ 492 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-7&0&7&0&6\\& & 18& 36& 58& 116& 246& \color{blue}{492} \\ \hline &9&18&29&58&123&\color{blue}{246}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 492 } = \color{orangered}{ 498 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-7&0&7&0&\color{orangered}{ 6 }\\& & 18& 36& 58& 116& 246& \color{orangered}{492} \\ \hline &\color{blue}{9}&\color{blue}{18}&\color{blue}{29}&\color{blue}{58}&\color{blue}{123}&\color{blue}{246}&\color{orangered}{498} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{5}+18x^{4}+29x^{3}+58x^{2}+123x+246 } $ with a remainder of $ \color{red}{ 498 } $.