Divide using synthetic division $$ ( 4x^{3}-11x+5 ) \div ( x-0.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&4&0&-11&5\\& & 2& 1& \color{black}{-5} \\ \hline &\color{blue}{4}&\color{blue}{2}&\color{blue}{-10}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-11x+5 }{ x-\frac{ 1 }{ 2 } } = \color{blue}{4x^{2}+2x-10} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-11&5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&\color{orangered}{ 4 }&0&-11&5\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$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}{ 4 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-11&5\\& & \color{blue}{2} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&4&\color{orangered}{ 0 }&-11&5\\& & \color{orangered}{2} & & \\ \hline &4&\color{orangered}{2}&& \end{array} $$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}{ 2 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-11&5\\& & 2& \color{blue}{1} & \\ \hline &4&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 1 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&4&0&\color{orangered}{ -11 }&5\\& & 2& \color{orangered}{1} & \\ \hline &4&2&\color{orangered}{-10}& \end{array} $$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( -10 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-11&5\\& & 2& 1& \color{blue}{-5} \\ \hline &4&2&\color{blue}{-10}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&4&0&-11&\color{orangered}{ 5 }\\& & 2& 1& \color{orangered}{-5} \\ \hline &\color{blue}{4}&\color{blue}{2}&\color{blue}{-10}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}+2x-10 } $ with a remainder of $ \color{red}{ 0 } $.