Divide using synthetic division $$ ( x^{4}+x^{3}-11x^{2}-5x+30 ) \div ( 1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&1&1&-11&-5&30\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-11}&\color{blue}{-5}&\color{orangered}{30} \end{array} $$The solution is:
$$ \frac{ x^{4}+x^{3}-11x^{2}-5x+30 }{ x } = \color{blue}{x^{3}+x^{2}-11x-5} ~+~ \frac{ \color{red}{ 30 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&1&-11&-5&30\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 1 }&1&-11&-5&30\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&1&-11&-5&30\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}0&1&\color{orangered}{ 1 }&-11&-5&30\\& & \color{orangered}{0} & & & \\ \hline &1&\color{orangered}{1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&1&-11&-5&30\\& & 0& \color{blue}{0} & & \\ \hline &1&\color{blue}{1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 0 } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrr}0&1&1&\color{orangered}{ -11 }&-5&30\\& & 0& \color{orangered}{0} & & \\ \hline &1&1&\color{orangered}{-11}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&1&-11&-5&30\\& & 0& 0& \color{blue}{0} & \\ \hline &1&1&\color{blue}{-11}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 0 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}0&1&1&-11&\color{orangered}{ -5 }&30\\& & 0& 0& \color{orangered}{0} & \\ \hline &1&1&-11&\color{orangered}{-5}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&1&-11&-5&30\\& & 0& 0& 0& \color{blue}{0} \\ \hline &1&1&-11&\color{blue}{-5}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ 0 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrrr}0&1&1&-11&-5&\color{orangered}{ 30 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-11}&\color{blue}{-5}&\color{orangered}{30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+x^{2}-11x-5 } $ with a remainder of $ \color{red}{ 30 } $.