Divide using synthetic division $$ ( 2x^{6}+4x^{5}+2x+5 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&2&4&0&0&0&2&5\\& & 4& 16& 32& 64& 128& \color{black}{260} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{16}&\color{blue}{32}&\color{blue}{64}&\color{blue}{130}&\color{orangered}{265} \end{array} $$The solution is:
$$ \frac{ 2x^{6}+4x^{5}+2x+5 }{ x-2 } = \color{blue}{2x^{5}+8x^{4}+16x^{3}+32x^{2}+64x+130} ~+~ \frac{ \color{red}{ 265 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&2&4&0&0&0&2&5\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 2 }&4&0&0&0&2&5\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&2&4&0&0&0&2&5\\& & \color{blue}{4} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 4 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}2&2&\color{orangered}{ 4 }&0&0&0&2&5\\& & \color{orangered}{4} & & & & & \\ \hline &2&\color{orangered}{8}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 8 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&2&4&0&0&0&2&5\\& & 4& \color{blue}{16} & & & & \\ \hline &2&\color{blue}{8}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrrr}2&2&4&\color{orangered}{ 0 }&0&0&2&5\\& & 4& \color{orangered}{16} & & & & \\ \hline &2&8&\color{orangered}{16}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 16 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&2&4&0&0&0&2&5\\& & 4& 16& \color{blue}{32} & & & \\ \hline &2&8&\color{blue}{16}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 32 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrrrr}2&2&4&0&\color{orangered}{ 0 }&0&2&5\\& & 4& 16& \color{orangered}{32} & & & \\ \hline &2&8&16&\color{orangered}{32}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 32 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&2&4&0&0&0&2&5\\& & 4& 16& 32& \color{blue}{64} & & \\ \hline &2&8&16&\color{blue}{32}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 64 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrrrr}2&2&4&0&0&\color{orangered}{ 0 }&2&5\\& & 4& 16& 32& \color{orangered}{64} & & \\ \hline &2&8&16&32&\color{orangered}{64}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 64 } = \color{blue}{ 128 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&2&4&0&0&0&2&5\\& & 4& 16& 32& 64& \color{blue}{128} & \\ \hline &2&8&16&32&\color{blue}{64}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 128 } = \color{orangered}{ 130 } $
$$ \begin{array}{c|rrrrrrr}2&2&4&0&0&0&\color{orangered}{ 2 }&5\\& & 4& 16& 32& 64& \color{orangered}{128} & \\ \hline &2&8&16&32&64&\color{orangered}{130}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 130 } = \color{blue}{ 260 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&2&4&0&0&0&2&5\\& & 4& 16& 32& 64& 128& \color{blue}{260} \\ \hline &2&8&16&32&64&\color{blue}{130}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 260 } = \color{orangered}{ 265 } $
$$ \begin{array}{c|rrrrrrr}2&2&4&0&0&0&2&\color{orangered}{ 5 }\\& & 4& 16& 32& 64& 128& \color{orangered}{260} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{16}&\color{blue}{32}&\color{blue}{64}&\color{blue}{130}&\color{orangered}{265} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{5}+8x^{4}+16x^{3}+32x^{2}+64x+130 } $ with a remainder of $ \color{red}{ 265 } $.