Divide using synthetic division $$ ( x^{4}-7x^{3}+x^{2}-7x ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-7&1&-7&1&-7&0\\& & -7& 98& -693& \color{black}{4900} \\ \hline &\color{blue}{1}&\color{blue}{-14}&\color{blue}{99}&\color{blue}{-700}&\color{orangered}{4900} \end{array} $$The solution is:
$$ \frac{ x^{4}-7x^{3}+x^{2}-7x }{ x+7 } = \color{blue}{x^{3}-14x^{2}+99x-700} ~+~ \frac{ \color{red}{ 4900 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-7&1&-7&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-7&\color{orangered}{ 1 }&-7&1&-7&0\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 1 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-7&1&-7&0\\& & \color{blue}{-7} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrrr}-7&1&\color{orangered}{ -7 }&1&-7&0\\& & \color{orangered}{-7} & & & \\ \hline &1&\color{orangered}{-14}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 98 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-7&1&-7&0\\& & -7& \color{blue}{98} & & \\ \hline &1&\color{blue}{-14}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 98 } = \color{orangered}{ 99 } $
$$ \begin{array}{c|rrrrr}-7&1&-7&\color{orangered}{ 1 }&-7&0\\& & -7& \color{orangered}{98} & & \\ \hline &1&-14&\color{orangered}{99}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 99 } = \color{blue}{ -693 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-7&1&-7&0\\& & -7& 98& \color{blue}{-693} & \\ \hline &1&-14&\color{blue}{99}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -693 \right) } = \color{orangered}{ -700 } $
$$ \begin{array}{c|rrrrr}-7&1&-7&1&\color{orangered}{ -7 }&0\\& & -7& 98& \color{orangered}{-693} & \\ \hline &1&-14&99&\color{orangered}{-700}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -700 \right) } = \color{blue}{ 4900 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-7&1&-7&0\\& & -7& 98& -693& \color{blue}{4900} \\ \hline &1&-14&99&\color{blue}{-700}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4900 } = \color{orangered}{ 4900 } $
$$ \begin{array}{c|rrrrr}-7&1&-7&1&-7&\color{orangered}{ 0 }\\& & -7& 98& -693& \color{orangered}{4900} \\ \hline &\color{blue}{1}&\color{blue}{-14}&\color{blue}{99}&\color{blue}{-700}&\color{orangered}{4900} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-14x^{2}+99x-700 } $ with a remainder of $ \color{red}{ 4900 } $.