Divide using synthetic division $$ ( 2x^{3}-x^{2}-5x-2 ) \div ( x-0.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&-1&-5&-2\\& & 1& 0& \color{black}{-\frac{ 5 }{ 2 }} \\ \hline &\color{blue}{2}&\color{blue}{0}&\color{blue}{-5}&\color{orangered}{-\frac{ 9 }{ 2 }} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-x^{2}-5x-2 }{ x-\frac{ 1 }{ 2 } } = \color{blue}{2x^{2}-5} \color{red}{~-~} \frac{ \color{red}{ \frac{ 9 }{ 2 } } }{ x-\frac{ 1 }{ 2 } } $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&-1&-5&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&\color{orangered}{ 2 }&-1&-5&-2\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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}{ 2 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&-1&-5&-2\\& & \color{blue}{1} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 1 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&\color{orangered}{ -1 }&-5&-2\\& & \color{orangered}{1} & & \\ \hline &2&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 1 }{ 2 } } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&-1&-5&-2\\& & 1& \color{blue}{0} & \\ \hline &2&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 0 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&-1&\color{orangered}{ -5 }&-2\\& & 1& \color{orangered}{0} & \\ \hline &2&0&\color{orangered}{-5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 1 }{ 2 } } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -\frac{ 5 }{ 2 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&-1&-5&-2\\& & 1& 0& \color{blue}{-\frac{ 5 }{ 2 }} \\ \hline &2&0&\color{blue}{-5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -\frac{ 5 }{ 2 } \right) } = \color{orangered}{ -\frac{ 9 }{ 2 } } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&-1&-5&\color{orangered}{ -2 }\\& & 1& 0& \color{orangered}{-\frac{ 5 }{ 2 }} \\ \hline &\color{blue}{2}&\color{blue}{0}&\color{blue}{-5}&\color{orangered}{-\frac{ 9 }{ 2 }} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-5 } $ with a remainder of $ \color{red}{ -\frac{ 9 }{ 2 } } $.