Divide using synthetic division $$ ( -13x^{2}+30x+56 ) \div ( -5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&-13&30&56\\& & 0& \color{black}{0} \\ \hline &\color{blue}{-13}&\color{blue}{30}&\color{orangered}{56} \end{array} $$The solution is:
$$ \frac{ -13x^{2}+30x+56 }{ x } = \color{blue}{-13x+30} ~+~ \frac{ \color{red}{ 56 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&-13&30&56\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ -13 }&30&56\\& & & \\ \hline &\color{orangered}{-13}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-13&30&56\\& & \color{blue}{0} & \\ \hline &\color{blue}{-13}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ 0 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrr}0&-13&\color{orangered}{ 30 }&56\\& & \color{orangered}{0} & \\ \hline &-13&\color{orangered}{30}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 30 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-13&30&56\\& & 0& \color{blue}{0} \\ \hline &-13&\color{blue}{30}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 56 } + \color{orangered}{ 0 } = \color{orangered}{ 56 } $
$$ \begin{array}{c|rrr}0&-13&30&\color{orangered}{ 56 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{-13}&\color{blue}{30}&\color{orangered}{56} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -13x+30 } $ with a remainder of $ \color{red}{ 56 } $.