Divide using synthetic division $$ ( 9x^{6}-9x^{4}+7x^{2}+8 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&9&0&-9&0&7&0&8\\& & 18& 36& 54& 108& 230& \color{black}{460} \\ \hline &\color{blue}{9}&\color{blue}{18}&\color{blue}{27}&\color{blue}{54}&\color{blue}{115}&\color{blue}{230}&\color{orangered}{468} \end{array} $$The solution is:
$$ \frac{ 9x^{6}-9x^{4}+7x^{2}+8 }{ x-2 } = \color{blue}{9x^{5}+18x^{4}+27x^{3}+54x^{2}+115x+230} ~+~ \frac{ \color{red}{ 468 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-9&0&7&0&8\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 9 }&0&-9&0&7&0&8\\& & & & & & & \\ \hline &\color{orangered}{9}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-9&0&7&0&8\\& & \color{blue}{18} & & & & & \\ \hline &\color{blue}{9}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrrr}2&9&\color{orangered}{ 0 }&-9&0&7&0&8\\& & \color{orangered}{18} & & & & & \\ \hline &9&\color{orangered}{18}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-9&0&7&0&8\\& & 18& \color{blue}{36} & & & & \\ \hline &9&\color{blue}{18}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 36 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&\color{orangered}{ -9 }&0&7&0&8\\& & 18& \color{orangered}{36} & & & & \\ \hline &9&18&\color{orangered}{27}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 27 } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-9&0&7&0&8\\& & 18& 36& \color{blue}{54} & & & \\ \hline &9&18&\color{blue}{27}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 54 } = \color{orangered}{ 54 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-9&\color{orangered}{ 0 }&7&0&8\\& & 18& 36& \color{orangered}{54} & & & \\ \hline &9&18&27&\color{orangered}{54}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 54 } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-9&0&7&0&8\\& & 18& 36& 54& \color{blue}{108} & & \\ \hline &9&18&27&\color{blue}{54}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 108 } = \color{orangered}{ 115 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-9&0&\color{orangered}{ 7 }&0&8\\& & 18& 36& 54& \color{orangered}{108} & & \\ \hline &9&18&27&54&\color{orangered}{115}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 115 } = \color{blue}{ 230 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-9&0&7&0&8\\& & 18& 36& 54& 108& \color{blue}{230} & \\ \hline &9&18&27&54&\color{blue}{115}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 230 } = \color{orangered}{ 230 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-9&0&7&\color{orangered}{ 0 }&8\\& & 18& 36& 54& 108& \color{orangered}{230} & \\ \hline &9&18&27&54&115&\color{orangered}{230}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 230 } = \color{blue}{ 460 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-9&0&7&0&8\\& & 18& 36& 54& 108& 230& \color{blue}{460} \\ \hline &9&18&27&54&115&\color{blue}{230}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 460 } = \color{orangered}{ 468 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-9&0&7&0&\color{orangered}{ 8 }\\& & 18& 36& 54& 108& 230& \color{orangered}{460} \\ \hline &\color{blue}{9}&\color{blue}{18}&\color{blue}{27}&\color{blue}{54}&\color{blue}{115}&\color{blue}{230}&\color{orangered}{468} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{5}+18x^{4}+27x^{3}+54x^{2}+115x+230 } $ with a remainder of $ \color{red}{ 468 } $.