Find remainder when $ 9x^{5}-9x^{4}-11x^{3}+6x+30 $ is divided by $ x+1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-1&9&-9&-11&0&6&30\\& & -9& 18& -7& 7& \color{black}{-13} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{7}&\color{blue}{-7}&\color{blue}{13}&\color{orangered}{17} \end{array} $$The remainder when $ 9x^{5}-9x^{4}-11x^{3}+6x+30 $ is divided by $ x+1 $ is $ \, \color{red}{ 17 } $.
Explanation
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&9&-9&-11&0&6&30\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-1&\color{orangered}{ 9 }&-9&-11&0&6&30\\& & & & & & \\ \hline &\color{orangered}{9}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 9 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&9&-9&-11&0&6&30\\& & \color{blue}{-9} & & & & \\ \hline &\color{blue}{9}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrrr}-1&9&\color{orangered}{ -9 }&-11&0&6&30\\& & \color{orangered}{-9} & & & & \\ \hline &9&\color{orangered}{-18}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&9&-9&-11&0&6&30\\& & -9& \color{blue}{18} & & & \\ \hline &9&\color{blue}{-18}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 18 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrr}-1&9&-9&\color{orangered}{ -11 }&0&6&30\\& & -9& \color{orangered}{18} & & & \\ \hline &9&-18&\color{orangered}{7}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 7 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&9&-9&-11&0&6&30\\& & -9& 18& \color{blue}{-7} & & \\ \hline &9&-18&\color{blue}{7}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrrr}-1&9&-9&-11&\color{orangered}{ 0 }&6&30\\& & -9& 18& \color{orangered}{-7} & & \\ \hline &9&-18&7&\color{orangered}{-7}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&9&-9&-11&0&6&30\\& & -9& 18& -7& \color{blue}{7} & \\ \hline &9&-18&7&\color{blue}{-7}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 7 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrrr}-1&9&-9&-11&0&\color{orangered}{ 6 }&30\\& & -9& 18& -7& \color{orangered}{7} & \\ \hline &9&-18&7&-7&\color{orangered}{13}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 13 } = \color{blue}{ -13 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&9&-9&-11&0&6&30\\& & -9& 18& -7& 7& \color{blue}{-13} \\ \hline &9&-18&7&-7&\color{blue}{13}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -13 \right) } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrrrr}-1&9&-9&-11&0&6&\color{orangered}{ 30 }\\& & -9& 18& -7& 7& \color{orangered}{-13} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{7}&\color{blue}{-7}&\color{blue}{13}&\color{orangered}{17} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 17 }\right) $.