Divide using synthetic division $$ ( x^{5}-6x^{4}+11x^{3}-x^{2}-14x+5 ) \div ( -x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&1&-6&11&-1&-14&5\\& & -2& 16& -54& 110& \color{black}{-192} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{27}&\color{blue}{-55}&\color{blue}{96}&\color{orangered}{-187} \end{array} $$The solution is:
$$ \frac{ x^{5}-6x^{4}+11x^{3}-x^{2}-14x+5 }{ x+2 } = \color{blue}{x^{4}-8x^{3}+27x^{2}-55x+96} \color{red}{~-~} \frac{ \color{red}{ 187 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&-6&11&-1&-14&5\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 1 }&-6&11&-1&-14&5\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&-6&11&-1&-14&5\\& & \color{blue}{-2} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrr}-2&1&\color{orangered}{ -6 }&11&-1&-14&5\\& & \color{orangered}{-2} & & & & \\ \hline &1&\color{orangered}{-8}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&-6&11&-1&-14&5\\& & -2& \color{blue}{16} & & & \\ \hline &1&\color{blue}{-8}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 16 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrrrr}-2&1&-6&\color{orangered}{ 11 }&-1&-14&5\\& & -2& \color{orangered}{16} & & & \\ \hline &1&-8&\color{orangered}{27}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 27 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&-6&11&-1&-14&5\\& & -2& 16& \color{blue}{-54} & & \\ \hline &1&-8&\color{blue}{27}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -55 } $
$$ \begin{array}{c|rrrrrr}-2&1&-6&11&\color{orangered}{ -1 }&-14&5\\& & -2& 16& \color{orangered}{-54} & & \\ \hline &1&-8&27&\color{orangered}{-55}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -55 \right) } = \color{blue}{ 110 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&-6&11&-1&-14&5\\& & -2& 16& -54& \color{blue}{110} & \\ \hline &1&-8&27&\color{blue}{-55}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 110 } = \color{orangered}{ 96 } $
$$ \begin{array}{c|rrrrrr}-2&1&-6&11&-1&\color{orangered}{ -14 }&5\\& & -2& 16& -54& \color{orangered}{110} & \\ \hline &1&-8&27&-55&\color{orangered}{96}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 96 } = \color{blue}{ -192 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&-6&11&-1&-14&5\\& & -2& 16& -54& 110& \color{blue}{-192} \\ \hline &1&-8&27&-55&\color{blue}{96}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -192 \right) } = \color{orangered}{ -187 } $
$$ \begin{array}{c|rrrrrr}-2&1&-6&11&-1&-14&\color{orangered}{ 5 }\\& & -2& 16& -54& 110& \color{orangered}{-192} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{27}&\color{blue}{-55}&\color{blue}{96}&\color{orangered}{-187} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-8x^{3}+27x^{2}-55x+96 } $ with a remainder of $ \color{red}{ -187 } $.