Divide using synthetic division $$ ( 6x+3 ) \div ( x-0.5 ) $$

The synthetic division table is:

$$ \begin{array}{c|rr}\frac{ 1 }{ 2 }&6&3\\& & \color{black}{3} \\ \hline &\color{blue}{6}&\color{orangered}{6} \end{array} $$

The solution is:

$$ \frac{ 6x+3 }{ x-\frac{ 1 }{ 2 } } = \color{blue}{6} ~+~ \frac{ \color{red}{ 6 } }{ x-\frac{ 1 }{ 2 } } $$

Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 1 }{ 2 } = 0 $ ( $ x = \color{blue}{ \frac{ 1 }{ 2 } } $ ) at the left.

$$ \begin{array}{c|rr}\color{blue}{\frac{ 1 }{ 2 }}&6&3\\& & \\ \hline && \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rr}\frac{ 1 }{ 2 }&\color{orangered}{ 6 }&3\\& & \\ \hline &\color{orangered}{6}& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 1 }{ 2 } } \cdot \color{blue}{ 6 } = \color{blue}{ 3 } $.

$$ \begin{array}{c|rr}\color{blue}{\frac{ 1 }{ 2 }}&6&3\\& & \color{blue}{3} \\ \hline &\color{blue}{6}& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 3 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rr}\frac{ 1 }{ 2 }&6&\color{orangered}{ 3 }\\& & \color{orangered}{3} \\ \hline &\color{blue}{6}&\color{orangered}{6} \end{array} $$

Bottom line represents the quotient $ \color{blue}{ 6 } $ with a remainder of $ \color{red}{ 6 } $.

Synthetic Division Calculator