Divide using synthetic division $$ ( 9x^{6}-5x^{4}+x^{2}+9 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&9&0&-5&0&1&0&9\\& & 18& 36& 62& 124& 250& \color{black}{500} \\ \hline &\color{blue}{9}&\color{blue}{18}&\color{blue}{31}&\color{blue}{62}&\color{blue}{125}&\color{blue}{250}&\color{orangered}{509} \end{array} $$The solution is:
$$ \frac{ 9x^{6}-5x^{4}+x^{2}+9 }{ x-2 } = \color{blue}{9x^{5}+18x^{4}+31x^{3}+62x^{2}+125x+250} ~+~ \frac{ \color{red}{ 509 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-5&0&1&0&9\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 9 }&0&-5&0&1&0&9\\& & & & & & & \\ \hline &\color{orangered}{9}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-5&0&1&0&9\\& & \color{blue}{18} & & & & & \\ \hline &\color{blue}{9}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrrr}2&9&\color{orangered}{ 0 }&-5&0&1&0&9\\& & \color{orangered}{18} & & & & & \\ \hline &9&\color{orangered}{18}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-5&0&1&0&9\\& & 18& \color{blue}{36} & & & & \\ \hline &9&\color{blue}{18}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 36 } = \color{orangered}{ 31 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&\color{orangered}{ -5 }&0&1&0&9\\& & 18& \color{orangered}{36} & & & & \\ \hline &9&18&\color{orangered}{31}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 31 } = \color{blue}{ 62 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-5&0&1&0&9\\& & 18& 36& \color{blue}{62} & & & \\ \hline &9&18&\color{blue}{31}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 62 } = \color{orangered}{ 62 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-5&\color{orangered}{ 0 }&1&0&9\\& & 18& 36& \color{orangered}{62} & & & \\ \hline &9&18&31&\color{orangered}{62}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 62 } = \color{blue}{ 124 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-5&0&1&0&9\\& & 18& 36& 62& \color{blue}{124} & & \\ \hline &9&18&31&\color{blue}{62}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 124 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-5&0&\color{orangered}{ 1 }&0&9\\& & 18& 36& 62& \color{orangered}{124} & & \\ \hline &9&18&31&62&\color{orangered}{125}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 125 } = \color{blue}{ 250 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-5&0&1&0&9\\& & 18& 36& 62& 124& \color{blue}{250} & \\ \hline &9&18&31&62&\color{blue}{125}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 250 } = \color{orangered}{ 250 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-5&0&1&\color{orangered}{ 0 }&9\\& & 18& 36& 62& 124& \color{orangered}{250} & \\ \hline &9&18&31&62&125&\color{orangered}{250}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 250 } = \color{blue}{ 500 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&9&0&-5&0&1&0&9\\& & 18& 36& 62& 124& 250& \color{blue}{500} \\ \hline &9&18&31&62&125&\color{blue}{250}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 500 } = \color{orangered}{ 509 } $
$$ \begin{array}{c|rrrrrrr}2&9&0&-5&0&1&0&\color{orangered}{ 9 }\\& & 18& 36& 62& 124& 250& \color{orangered}{500} \\ \hline &\color{blue}{9}&\color{blue}{18}&\color{blue}{31}&\color{blue}{62}&\color{blue}{125}&\color{blue}{250}&\color{orangered}{509} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{5}+18x^{4}+31x^{3}+62x^{2}+125x+250 } $ with a remainder of $ \color{red}{ 509 } $.