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