Divide using synthetic division $$ ( 5x^{6}-2x^{4}+8x^{2}+2 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&5&0&-2&0&8&0&2\\& & 10& 20& 36& 72& 160& \color{black}{320} \\ \hline &\color{blue}{5}&\color{blue}{10}&\color{blue}{18}&\color{blue}{36}&\color{blue}{80}&\color{blue}{160}&\color{orangered}{322} \end{array} $$The solution is:
$$ \frac{ 5x^{6}-2x^{4}+8x^{2}+2 }{ x-2 } = \color{blue}{5x^{5}+10x^{4}+18x^{3}+36x^{2}+80x+160} ~+~ \frac{ \color{red}{ 322 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&5&0&-2&0&8&0&2\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 5 }&0&-2&0&8&0&2\\& & & & & & & \\ \hline &\color{orangered}{5}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 5 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&5&0&-2&0&8&0&2\\& & \color{blue}{10} & & & & & \\ \hline &\color{blue}{5}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 10 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrrr}2&5&\color{orangered}{ 0 }&-2&0&8&0&2\\& & \color{orangered}{10} & & & & & \\ \hline &5&\color{orangered}{10}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 10 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&5&0&-2&0&8&0&2\\& & 10& \color{blue}{20} & & & & \\ \hline &5&\color{blue}{10}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 20 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrrr}2&5&0&\color{orangered}{ -2 }&0&8&0&2\\& & 10& \color{orangered}{20} & & & & \\ \hline &5&10&\color{orangered}{18}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 18 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&5&0&-2&0&8&0&2\\& & 10& 20& \color{blue}{36} & & & \\ \hline &5&10&\color{blue}{18}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 36 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrrrrrr}2&5&0&-2&\color{orangered}{ 0 }&8&0&2\\& & 10& 20& \color{orangered}{36} & & & \\ \hline &5&10&18&\color{orangered}{36}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 36 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&5&0&-2&0&8&0&2\\& & 10& 20& 36& \color{blue}{72} & & \\ \hline &5&10&18&\color{blue}{36}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 72 } = \color{orangered}{ 80 } $
$$ \begin{array}{c|rrrrrrr}2&5&0&-2&0&\color{orangered}{ 8 }&0&2\\& & 10& 20& 36& \color{orangered}{72} & & \\ \hline &5&10&18&36&\color{orangered}{80}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 80 } = \color{blue}{ 160 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&5&0&-2&0&8&0&2\\& & 10& 20& 36& 72& \color{blue}{160} & \\ \hline &5&10&18&36&\color{blue}{80}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 160 } = \color{orangered}{ 160 } $
$$ \begin{array}{c|rrrrrrr}2&5&0&-2&0&8&\color{orangered}{ 0 }&2\\& & 10& 20& 36& 72& \color{orangered}{160} & \\ \hline &5&10&18&36&80&\color{orangered}{160}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 160 } = \color{blue}{ 320 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&5&0&-2&0&8&0&2\\& & 10& 20& 36& 72& 160& \color{blue}{320} \\ \hline &5&10&18&36&80&\color{blue}{160}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 320 } = \color{orangered}{ 322 } $
$$ \begin{array}{c|rrrrrrr}2&5&0&-2&0&8&0&\color{orangered}{ 2 }\\& & 10& 20& 36& 72& 160& \color{orangered}{320} \\ \hline &\color{blue}{5}&\color{blue}{10}&\color{blue}{18}&\color{blue}{36}&\color{blue}{80}&\color{blue}{160}&\color{orangered}{322} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{5}+10x^{4}+18x^{3}+36x^{2}+80x+160 } $ with a remainder of $ \color{red}{ 322 } $.