Divide using synthetic division $$ ( -x^{5}+6x^{4}-11x^{3}+5 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-1&-1&6&-11&0&0&5\\& & 1& -7& 18& -18& \color{black}{18} \\ \hline &\color{blue}{-1}&\color{blue}{7}&\color{blue}{-18}&\color{blue}{18}&\color{blue}{-18}&\color{orangered}{23} \end{array} $$The solution is:
$$ \frac{ -x^{5}+6x^{4}-11x^{3}+5 }{ x+1 } = \color{blue}{-x^{4}+7x^{3}-18x^{2}+18x-18} ~+~ \frac{ \color{red}{ 23 } }{ x+1 } $$Explanation
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&6&-11&0&0&5\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-1&\color{orangered}{ -1 }&6&-11&0&0&5\\& & & & & & \\ \hline &\color{orangered}{-1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&6&-11&0&0&5\\& & \color{blue}{1} & & & & \\ \hline &\color{blue}{-1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 1 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrr}-1&-1&\color{orangered}{ 6 }&-11&0&0&5\\& & \color{orangered}{1} & & & & \\ \hline &-1&\color{orangered}{7}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 7 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&6&-11&0&0&5\\& & 1& \color{blue}{-7} & & & \\ \hline &-1&\color{blue}{7}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrrr}-1&-1&6&\color{orangered}{ -11 }&0&0&5\\& & 1& \color{orangered}{-7} & & & \\ \hline &-1&7&\color{orangered}{-18}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&6&-11&0&0&5\\& & 1& -7& \color{blue}{18} & & \\ \hline &-1&7&\color{blue}{-18}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}-1&-1&6&-11&\color{orangered}{ 0 }&0&5\\& & 1& -7& \color{orangered}{18} & & \\ \hline &-1&7&-18&\color{orangered}{18}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 18 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&6&-11&0&0&5\\& & 1& -7& 18& \color{blue}{-18} & \\ \hline &-1&7&-18&\color{blue}{18}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrrr}-1&-1&6&-11&0&\color{orangered}{ 0 }&5\\& & 1& -7& 18& \color{orangered}{-18} & \\ \hline &-1&7&-18&18&\color{orangered}{-18}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&6&-11&0&0&5\\& & 1& -7& 18& -18& \color{blue}{18} \\ \hline &-1&7&-18&18&\color{blue}{-18}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 18 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrrrr}-1&-1&6&-11&0&0&\color{orangered}{ 5 }\\& & 1& -7& 18& -18& \color{orangered}{18} \\ \hline &\color{blue}{-1}&\color{blue}{7}&\color{blue}{-18}&\color{blue}{18}&\color{blue}{-18}&\color{orangered}{23} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{4}+7x^{3}-18x^{2}+18x-18 } $ with a remainder of $ \color{red}{ 23 } $.