Divide using synthetic division $$ ( 2x^{3}+x^{2}-13x+9 ) \div ( x-0.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&1&-13&9\\& & 1& 1& \color{black}{-6} \\ \hline &\color{blue}{2}&\color{blue}{2}&\color{blue}{-12}&\color{orangered}{3} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+x^{2}-13x+9 }{ x-\frac{ 1 }{ 2 } } = \color{blue}{2x^{2}+2x-12} ~+~ \frac{ \color{red}{ 3 } }{ x-\frac{ 1 }{ 2 } } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&1&-13&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&\color{orangered}{ 2 }&1&-13&9\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&1&-13&9\\& & \color{blue}{1} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 1 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&\color{orangered}{ 1 }&-13&9\\& & \color{orangered}{1} & & \\ \hline &2&\color{orangered}{2}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&1&-13&9\\& & 1& \color{blue}{1} & \\ \hline &2&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 1 } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&1&\color{orangered}{ -13 }&9\\& & 1& \color{orangered}{1} & \\ \hline &2&2&\color{orangered}{-12}& \end{array} $$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( -12 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 1 }{ 2 }}&2&1&-13&9\\& & 1& 1& \color{blue}{-6} \\ \hline &2&2&\color{blue}{-12}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}\frac{ 1 }{ 2 }&2&1&-13&\color{orangered}{ 9 }\\& & 1& 1& \color{orangered}{-6} \\ \hline &\color{blue}{2}&\color{blue}{2}&\color{blue}{-12}&\color{orangered}{3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+2x-12 } $ with a remainder of $ \color{red}{ 3 } $.