Divide using synthetic division $$ ( x^{5}-1 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}5&1&0&0&0&0&-1\\& & 5& 25& 125& 625& \color{black}{3125} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{25}&\color{blue}{125}&\color{blue}{625}&\color{orangered}{3124} \end{array} $$The solution is:
$$ \frac{ x^{5}-1 }{ x-5 } = \color{blue}{x^{4}+5x^{3}+25x^{2}+125x+625} ~+~ \frac{ \color{red}{ 3124 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&1&0&0&0&0&-1\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}5&\color{orangered}{ 1 }&0&0&0&0&-1\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&1&0&0&0&0&-1\\& & \color{blue}{5} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}5&1&\color{orangered}{ 0 }&0&0&0&-1\\& & \color{orangered}{5} & & & & \\ \hline &1&\color{orangered}{5}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&1&0&0&0&0&-1\\& & 5& \color{blue}{25} & & & \\ \hline &1&\color{blue}{5}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 25 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrrr}5&1&0&\color{orangered}{ 0 }&0&0&-1\\& & 5& \color{orangered}{25} & & & \\ \hline &1&5&\color{orangered}{25}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 25 } = \color{blue}{ 125 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&1&0&0&0&0&-1\\& & 5& 25& \color{blue}{125} & & \\ \hline &1&5&\color{blue}{25}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 125 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrrrrr}5&1&0&0&\color{orangered}{ 0 }&0&-1\\& & 5& 25& \color{orangered}{125} & & \\ \hline &1&5&25&\color{orangered}{125}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 125 } = \color{blue}{ 625 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&1&0&0&0&0&-1\\& & 5& 25& 125& \color{blue}{625} & \\ \hline &1&5&25&\color{blue}{125}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 625 } = \color{orangered}{ 625 } $
$$ \begin{array}{c|rrrrrr}5&1&0&0&0&\color{orangered}{ 0 }&-1\\& & 5& 25& 125& \color{orangered}{625} & \\ \hline &1&5&25&125&\color{orangered}{625}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 625 } = \color{blue}{ 3125 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&1&0&0&0&0&-1\\& & 5& 25& 125& 625& \color{blue}{3125} \\ \hline &1&5&25&125&\color{blue}{625}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 3125 } = \color{orangered}{ 3124 } $
$$ \begin{array}{c|rrrrrr}5&1&0&0&0&0&\color{orangered}{ -1 }\\& & 5& 25& 125& 625& \color{orangered}{3125} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{25}&\color{blue}{125}&\color{blue}{625}&\color{orangered}{3124} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+5x^{3}+25x^{2}+125x+625 } $ with a remainder of $ \color{red}{ 3124 } $.