Divide using synthetic division $$ ( -x^{3}+2x^{2}+19x-20 ) \div ( 0 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-1&2&19&-20\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{19}&\color{orangered}{-20} \end{array} $$The solution is:
$$ \frac{ -x^{3}+2x^{2}+19x-20 }{ x } = \color{blue}{-x^{2}+2x+19} \color{red}{~-~} \frac{ \color{red}{ 20 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&2&19&-20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -1 }&2&19&-20\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&2&19&-20\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}0&-1&\color{orangered}{ 2 }&19&-20\\& & \color{orangered}{0} & & \\ \hline &-1&\color{orangered}{2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 2 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&2&19&-20\\& & 0& \color{blue}{0} & \\ \hline &-1&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ 0 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}0&-1&2&\color{orangered}{ 19 }&-20\\& & 0& \color{orangered}{0} & \\ \hline &-1&2&\color{orangered}{19}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-1&2&19&-20\\& & 0& 0& \color{blue}{0} \\ \hline &-1&2&\color{blue}{19}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 0 } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrr}0&-1&2&19&\color{orangered}{ -20 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{19}&\color{orangered}{-20} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+2x+19 } $ with a remainder of $ \color{red}{ -20 } $.