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