Divide using synthetic division $$ ( -x^{3}+19x+30 ) \div ( x ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-1&0&19&30\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-1}&\color{blue}{0}&\color{blue}{19}&\color{orangered}{30} \end{array} $$The solution is:
$$ \frac{ -x^{3}+19x+30 }{ x } = \color{blue}{-x^{2}+19} ~+~ \frac{ \color{red}{ 30 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&0&19&30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -1 }&0&19&30\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&0&19&30\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}0&-1&\color{orangered}{ 0 }&19&30\\& & \color{orangered}{0} & & \\ \hline &-1&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&0&19&30\\& & 0& \color{blue}{0} & \\ \hline &-1&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ 0 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}0&-1&0&\color{orangered}{ 19 }&30\\& & 0& \color{orangered}{0} & \\ \hline &-1&0&\color{orangered}{19}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 19 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&0&19&30\\& & 0& 0& \color{blue}{0} \\ \hline &-1&0&\color{blue}{19}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ 0 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrr}0&-1&0&19&\color{orangered}{ 30 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-1}&\color{blue}{0}&\color{blue}{19}&\color{orangered}{30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+19 } $ with a remainder of $ \color{red}{ 30 } $.