Divide using synthetic division $$ ( 4x^{4}+11x^{3}+17x^{2}+x-9 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&4&11&17&1&-9\\& & -4& -7& -10& \color{black}{9} \\ \hline &\color{blue}{4}&\color{blue}{7}&\color{blue}{10}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 4x^{4}+11x^{3}+17x^{2}+x-9 }{ x+1 } = \color{blue}{4x^{3}+7x^{2}+10x-9} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&11&17&1&-9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 4 }&11&17&1&-9\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&11&17&1&-9\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}-1&4&\color{orangered}{ 11 }&17&1&-9\\& & \color{orangered}{-4} & & & \\ \hline &4&\color{orangered}{7}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&11&17&1&-9\\& & -4& \color{blue}{-7} & & \\ \hline &4&\color{blue}{7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrr}-1&4&11&\color{orangered}{ 17 }&1&-9\\& & -4& \color{orangered}{-7} & & \\ \hline &4&7&\color{orangered}{10}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 10 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&11&17&1&-9\\& & -4& -7& \color{blue}{-10} & \\ \hline &4&7&\color{blue}{10}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-1&4&11&17&\color{orangered}{ 1 }&-9\\& & -4& -7& \color{orangered}{-10} & \\ \hline &4&7&10&\color{orangered}{-9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&11&17&1&-9\\& & -4& -7& -10& \color{blue}{9} \\ \hline &4&7&10&\color{blue}{-9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 9 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-1&4&11&17&1&\color{orangered}{ -9 }\\& & -4& -7& -10& \color{orangered}{9} \\ \hline &\color{blue}{4}&\color{blue}{7}&\color{blue}{10}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}+7x^{2}+10x-9 } $ with a remainder of $ \color{red}{ 0 } $.