Divide using synthetic division $$ ( -11x^{2}+17x-6 ) \div ( -2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&-11&17&-6\\& & 0& \color{black}{0} \\ \hline &\color{blue}{-11}&\color{blue}{17}&\color{orangered}{-6} \end{array} $$The solution is:
$$ \frac{ -11x^{2}+17x-6 }{ x } = \color{blue}{-11x+17} \color{red}{~-~} \frac{ \color{red}{ 6 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&-11&17&-6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ -11 }&17&-6\\& & & \\ \hline &\color{orangered}{-11}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-11&17&-6\\& & \color{blue}{0} & \\ \hline &\color{blue}{-11}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 0 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrr}0&-11&\color{orangered}{ 17 }&-6\\& & \color{orangered}{0} & \\ \hline &-11&\color{orangered}{17}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 17 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-11&17&-6\\& & 0& \color{blue}{0} \\ \hline &-11&\color{blue}{17}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 0 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrr}0&-11&17&\color{orangered}{ -6 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{-11}&\color{blue}{17}&\color{orangered}{-6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -11x+17 } $ with a remainder of $ \color{red}{ -6 } $.