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