Divide using synthetic division $$ ( 8x^{6}-8x^{4}+2x^{2}+5 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&8&0&-8&0&2&0&5\\& & 16& 32& 48& 96& 196& \color{black}{392} \\ \hline &\color{blue}{8}&\color{blue}{16}&\color{blue}{24}&\color{blue}{48}&\color{blue}{98}&\color{blue}{196}&\color{orangered}{397} \end{array} $$The solution is:
$$ \frac{ 8x^{6}-8x^{4}+2x^{2}+5 }{ x-2 } = \color{blue}{8x^{5}+16x^{4}+24x^{3}+48x^{2}+98x+196} ~+~ \frac{ \color{red}{ 397 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-8&0&2&0&5\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 8 }&0&-8&0&2&0&5\\& & & & & & & \\ \hline &\color{orangered}{8}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 8 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-8&0&2&0&5\\& & \color{blue}{16} & & & & & \\ \hline &\color{blue}{8}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrrr}2&8&\color{orangered}{ 0 }&-8&0&2&0&5\\& & \color{orangered}{16} & & & & & \\ \hline &8&\color{orangered}{16}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 16 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-8&0&2&0&5\\& & 16& \color{blue}{32} & & & & \\ \hline &8&\color{blue}{16}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 32 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&\color{orangered}{ -8 }&0&2&0&5\\& & 16& \color{orangered}{32} & & & & \\ \hline &8&16&\color{orangered}{24}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 24 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-8&0&2&0&5\\& & 16& 32& \color{blue}{48} & & & \\ \hline &8&16&\color{blue}{24}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 48 } = \color{orangered}{ 48 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-8&\color{orangered}{ 0 }&2&0&5\\& & 16& 32& \color{orangered}{48} & & & \\ \hline &8&16&24&\color{orangered}{48}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 48 } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-8&0&2&0&5\\& & 16& 32& 48& \color{blue}{96} & & \\ \hline &8&16&24&\color{blue}{48}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 96 } = \color{orangered}{ 98 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-8&0&\color{orangered}{ 2 }&0&5\\& & 16& 32& 48& \color{orangered}{96} & & \\ \hline &8&16&24&48&\color{orangered}{98}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 98 } = \color{blue}{ 196 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-8&0&2&0&5\\& & 16& 32& 48& 96& \color{blue}{196} & \\ \hline &8&16&24&48&\color{blue}{98}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 196 } = \color{orangered}{ 196 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-8&0&2&\color{orangered}{ 0 }&5\\& & 16& 32& 48& 96& \color{orangered}{196} & \\ \hline &8&16&24&48&98&\color{orangered}{196}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 196 } = \color{blue}{ 392 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-8&0&2&0&5\\& & 16& 32& 48& 96& 196& \color{blue}{392} \\ \hline &8&16&24&48&98&\color{blue}{196}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 392 } = \color{orangered}{ 397 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-8&0&2&0&\color{orangered}{ 5 }\\& & 16& 32& 48& 96& 196& \color{orangered}{392} \\ \hline &\color{blue}{8}&\color{blue}{16}&\color{blue}{24}&\color{blue}{48}&\color{blue}{98}&\color{blue}{196}&\color{orangered}{397} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{5}+16x^{4}+24x^{3}+48x^{2}+98x+196 } $ with a remainder of $ \color{red}{ 397 } $.