Divide using synthetic division $$ ( 6x^{6}-x^{4}+9x^{2}+7 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&6&0&-1&0&9&0&7\\& & 12& 24& 46& 92& 202& \color{black}{404} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{23}&\color{blue}{46}&\color{blue}{101}&\color{blue}{202}&\color{orangered}{411} \end{array} $$The solution is:
$$ \frac{ 6x^{6}-x^{4}+9x^{2}+7 }{ x-2 } = \color{blue}{6x^{5}+12x^{4}+23x^{3}+46x^{2}+101x+202} ~+~ \frac{ \color{red}{ 411 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-1&0&9&0&7\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 6 }&0&-1&0&9&0&7\\& & & & & & & \\ \hline &\color{orangered}{6}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-1&0&9&0&7\\& & \color{blue}{12} & & & & & \\ \hline &\color{blue}{6}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrrr}2&6&\color{orangered}{ 0 }&-1&0&9&0&7\\& & \color{orangered}{12} & & & & & \\ \hline &6&\color{orangered}{12}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 12 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-1&0&9&0&7\\& & 12& \color{blue}{24} & & & & \\ \hline &6&\color{blue}{12}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 24 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&\color{orangered}{ -1 }&0&9&0&7\\& & 12& \color{orangered}{24} & & & & \\ \hline &6&12&\color{orangered}{23}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 23 } = \color{blue}{ 46 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-1&0&9&0&7\\& & 12& 24& \color{blue}{46} & & & \\ \hline &6&12&\color{blue}{23}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 46 } = \color{orangered}{ 46 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-1&\color{orangered}{ 0 }&9&0&7\\& & 12& 24& \color{orangered}{46} & & & \\ \hline &6&12&23&\color{orangered}{46}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 46 } = \color{blue}{ 92 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-1&0&9&0&7\\& & 12& 24& 46& \color{blue}{92} & & \\ \hline &6&12&23&\color{blue}{46}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 92 } = \color{orangered}{ 101 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-1&0&\color{orangered}{ 9 }&0&7\\& & 12& 24& 46& \color{orangered}{92} & & \\ \hline &6&12&23&46&\color{orangered}{101}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 101 } = \color{blue}{ 202 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-1&0&9&0&7\\& & 12& 24& 46& 92& \color{blue}{202} & \\ \hline &6&12&23&46&\color{blue}{101}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 202 } = \color{orangered}{ 202 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-1&0&9&\color{orangered}{ 0 }&7\\& & 12& 24& 46& 92& \color{orangered}{202} & \\ \hline &6&12&23&46&101&\color{orangered}{202}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 202 } = \color{blue}{ 404 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-1&0&9&0&7\\& & 12& 24& 46& 92& 202& \color{blue}{404} \\ \hline &6&12&23&46&101&\color{blue}{202}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 404 } = \color{orangered}{ 411 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-1&0&9&0&\color{orangered}{ 7 }\\& & 12& 24& 46& 92& 202& \color{orangered}{404} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{23}&\color{blue}{46}&\color{blue}{101}&\color{blue}{202}&\color{orangered}{411} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{5}+12x^{4}+23x^{3}+46x^{2}+101x+202 } $ with a remainder of $ \color{red}{ 411 } $.