Divide using synthetic division $$ ( x^{4}+2x^{3}-11x^{2}-7x+19 ) \div ( -4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&1&2&-11&-7&19\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-11}&\color{blue}{-7}&\color{orangered}{19} \end{array} $$The solution is:
$$ \frac{ x^{4}+2x^{3}-11x^{2}-7x+19 }{ x } = \color{blue}{x^{3}+2x^{2}-11x-7} ~+~ \frac{ \color{red}{ 19 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&2&-11&-7&19\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 1 }&2&-11&-7&19\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&2&-11&-7&19\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}0&1&\color{orangered}{ 2 }&-11&-7&19\\& & \color{orangered}{0} & & & \\ \hline &1&\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&2&-11&-7&19\\& & 0& \color{blue}{0} & & \\ \hline &1&\color{blue}{2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 0 } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrr}0&1&2&\color{orangered}{ -11 }&-7&19\\& & 0& \color{orangered}{0} & & \\ \hline &1&2&\color{orangered}{-11}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&2&-11&-7&19\\& & 0& 0& \color{blue}{0} & \\ \hline &1&2&\color{blue}{-11}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 0 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}0&1&2&-11&\color{orangered}{ -7 }&19\\& & 0& 0& \color{orangered}{0} & \\ \hline &1&2&-11&\color{orangered}{-7}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&2&-11&-7&19\\& & 0& 0& 0& \color{blue}{0} \\ \hline &1&2&-11&\color{blue}{-7}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ 0 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}0&1&2&-11&-7&\color{orangered}{ 19 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-11}&\color{blue}{-7}&\color{orangered}{19} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+2x^{2}-11x-7 } $ with a remainder of $ \color{red}{ 19 } $.