Divide using synthetic division $$ ( x^{5}+1 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&1&0&0&0&0&1\\& & 2& 4& 8& 16& \color{black}{32} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{4}&\color{blue}{8}&\color{blue}{16}&\color{orangered}{33} \end{array} $$The solution is:
$$ \frac{ x^{5}+1 }{ x-2 } = \color{blue}{x^{4}+2x^{3}+4x^{2}+8x+16} ~+~ \frac{ \color{red}{ 33 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&0&0&0&0&1\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 1 }&0&0&0&0&1\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&0&0&0&0&1\\& & \color{blue}{2} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}2&1&\color{orangered}{ 0 }&0&0&0&1\\& & \color{orangered}{2} & & & & \\ \hline &1&\color{orangered}{2}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&0&0&0&0&1\\& & 2& \color{blue}{4} & & & \\ \hline &1&\color{blue}{2}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}2&1&0&\color{orangered}{ 0 }&0&0&1\\& & 2& \color{orangered}{4} & & & \\ \hline &1&2&\color{orangered}{4}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&0&0&0&0&1\\& & 2& 4& \color{blue}{8} & & \\ \hline &1&2&\color{blue}{4}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}2&1&0&0&\color{orangered}{ 0 }&0&1\\& & 2& 4& \color{orangered}{8} & & \\ \hline &1&2&4&\color{orangered}{8}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 8 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&0&0&0&0&1\\& & 2& 4& 8& \color{blue}{16} & \\ \hline &1&2&4&\color{blue}{8}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrr}2&1&0&0&0&\color{orangered}{ 0 }&1\\& & 2& 4& 8& \color{orangered}{16} & \\ \hline &1&2&4&8&\color{orangered}{16}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 16 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&0&0&0&0&1\\& & 2& 4& 8& 16& \color{blue}{32} \\ \hline &1&2&4&8&\color{blue}{16}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 32 } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrrrrr}2&1&0&0&0&0&\color{orangered}{ 1 }\\& & 2& 4& 8& 16& \color{orangered}{32} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{4}&\color{blue}{8}&\color{blue}{16}&\color{orangered}{33} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+2x^{3}+4x^{2}+8x+16 } $ with a remainder of $ \color{red}{ 33 } $.