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