Divide using synthetic division $$ ( 8x^{6}-2x^{4}+5x^{2}+7 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&8&0&-2&0&5&0&7\\& & 16& 32& 60& 120& 250& \color{black}{500} \\ \hline &\color{blue}{8}&\color{blue}{16}&\color{blue}{30}&\color{blue}{60}&\color{blue}{125}&\color{blue}{250}&\color{orangered}{507} \end{array} $$The solution is:
$$ \frac{ 8x^{6}-2x^{4}+5x^{2}+7 }{ x-2 } = \color{blue}{8x^{5}+16x^{4}+30x^{3}+60x^{2}+125x+250} ~+~ \frac{ \color{red}{ 507 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-2&0&5&0&7\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 8 }&0&-2&0&5&0&7\\& & & & & & & \\ \hline &\color{orangered}{8}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-2&0&5&0&7\\& & \color{blue}{16} & & & & & \\ \hline &\color{blue}{8}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrrr}2&8&\color{orangered}{ 0 }&-2&0&5&0&7\\& & \color{orangered}{16} & & & & & \\ \hline &8&\color{orangered}{16}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-2&0&5&0&7\\& & 16& \color{blue}{32} & & & & \\ \hline &8&\color{blue}{16}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 32 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&\color{orangered}{ -2 }&0&5&0&7\\& & 16& \color{orangered}{32} & & & & \\ \hline &8&16&\color{orangered}{30}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 30 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-2&0&5&0&7\\& & 16& 32& \color{blue}{60} & & & \\ \hline &8&16&\color{blue}{30}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 60 } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-2&\color{orangered}{ 0 }&5&0&7\\& & 16& 32& \color{orangered}{60} & & & \\ \hline &8&16&30&\color{orangered}{60}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 60 } = \color{blue}{ 120 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-2&0&5&0&7\\& & 16& 32& 60& \color{blue}{120} & & \\ \hline &8&16&30&\color{blue}{60}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 120 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-2&0&\color{orangered}{ 5 }&0&7\\& & 16& 32& 60& \color{orangered}{120} & & \\ \hline &8&16&30&60&\color{orangered}{125}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-2&0&5&0&7\\& & 16& 32& 60& 120& \color{blue}{250} & \\ \hline &8&16&30&60&\color{blue}{125}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 250 } = \color{orangered}{ 250 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-2&0&5&\color{orangered}{ 0 }&7\\& & 16& 32& 60& 120& \color{orangered}{250} & \\ \hline &8&16&30&60&125&\color{orangered}{250}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&8&0&-2&0&5&0&7\\& & 16& 32& 60& 120& 250& \color{blue}{500} \\ \hline &8&16&30&60&125&\color{blue}{250}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 500 } = \color{orangered}{ 507 } $
$$ \begin{array}{c|rrrrrrr}2&8&0&-2&0&5&0&\color{orangered}{ 7 }\\& & 16& 32& 60& 120& 250& \color{orangered}{500} \\ \hline &\color{blue}{8}&\color{blue}{16}&\color{blue}{30}&\color{blue}{60}&\color{blue}{125}&\color{blue}{250}&\color{orangered}{507} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{5}+16x^{4}+30x^{3}+60x^{2}+125x+250 } $ with a remainder of $ \color{red}{ 507 } $.