Divide using synthetic division $$ ( 7x^{6}-9x^{4}+7x^{2}+7 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&7&0&-9&0&7&0&7\\& & 14& 28& 38& 76& 166& \color{black}{332} \\ \hline &\color{blue}{7}&\color{blue}{14}&\color{blue}{19}&\color{blue}{38}&\color{blue}{83}&\color{blue}{166}&\color{orangered}{339} \end{array} $$The solution is:
$$ \frac{ 7x^{6}-9x^{4}+7x^{2}+7 }{ x-2 } = \color{blue}{7x^{5}+14x^{4}+19x^{3}+38x^{2}+83x+166} ~+~ \frac{ \color{red}{ 339 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&7&0&-9&0&7&0&7\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 7 }&0&-9&0&7&0&7\\& & & & & & & \\ \hline &\color{orangered}{7}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 7 } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&7&0&-9&0&7&0&7\\& & \color{blue}{14} & & & & & \\ \hline &\color{blue}{7}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 14 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrrrr}2&7&\color{orangered}{ 0 }&-9&0&7&0&7\\& & \color{orangered}{14} & & & & & \\ \hline &7&\color{orangered}{14}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 14 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&7&0&-9&0&7&0&7\\& & 14& \color{blue}{28} & & & & \\ \hline &7&\color{blue}{14}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 28 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrrr}2&7&0&\color{orangered}{ -9 }&0&7&0&7\\& & 14& \color{orangered}{28} & & & & \\ \hline &7&14&\color{orangered}{19}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 19 } = \color{blue}{ 38 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&7&0&-9&0&7&0&7\\& & 14& 28& \color{blue}{38} & & & \\ \hline &7&14&\color{blue}{19}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 38 } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrrrrrr}2&7&0&-9&\color{orangered}{ 0 }&7&0&7\\& & 14& 28& \color{orangered}{38} & & & \\ \hline &7&14&19&\color{orangered}{38}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 38 } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&7&0&-9&0&7&0&7\\& & 14& 28& 38& \color{blue}{76} & & \\ \hline &7&14&19&\color{blue}{38}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 76 } = \color{orangered}{ 83 } $
$$ \begin{array}{c|rrrrrrr}2&7&0&-9&0&\color{orangered}{ 7 }&0&7\\& & 14& 28& 38& \color{orangered}{76} & & \\ \hline &7&14&19&38&\color{orangered}{83}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 83 } = \color{blue}{ 166 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&7&0&-9&0&7&0&7\\& & 14& 28& 38& 76& \color{blue}{166} & \\ \hline &7&14&19&38&\color{blue}{83}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 166 } = \color{orangered}{ 166 } $
$$ \begin{array}{c|rrrrrrr}2&7&0&-9&0&7&\color{orangered}{ 0 }&7\\& & 14& 28& 38& 76& \color{orangered}{166} & \\ \hline &7&14&19&38&83&\color{orangered}{166}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 166 } = \color{blue}{ 332 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&7&0&-9&0&7&0&7\\& & 14& 28& 38& 76& 166& \color{blue}{332} \\ \hline &7&14&19&38&83&\color{blue}{166}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 332 } = \color{orangered}{ 339 } $
$$ \begin{array}{c|rrrrrrr}2&7&0&-9&0&7&0&\color{orangered}{ 7 }\\& & 14& 28& 38& 76& 166& \color{orangered}{332} \\ \hline &\color{blue}{7}&\color{blue}{14}&\color{blue}{19}&\color{blue}{38}&\color{blue}{83}&\color{blue}{166}&\color{orangered}{339} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x^{5}+14x^{4}+19x^{3}+38x^{2}+83x+166 } $ with a remainder of $ \color{red}{ 339 } $.