Divide using synthetic division $$ ( x^{3}+2x^{2}-34x+9 ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&1&2&-34&9\\& & -7& 35& \color{black}{-7} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{1}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ x^{3}+2x^{2}-34x+9 }{ x+7 } = \color{blue}{x^{2}-5x+1} ~+~ \frac{ \color{red}{ 2 } }{ x+7 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&2&-34&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ 1 }&2&-34&9\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&2&-34&9\\& & \color{blue}{-7} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-7&1&\color{orangered}{ 2 }&-34&9\\& & \color{orangered}{-7} & & \\ \hline &1&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&2&-34&9\\& & -7& \color{blue}{35} & \\ \hline &1&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 35 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}-7&1&2&\color{orangered}{ -34 }&9\\& & -7& \color{orangered}{35} & \\ \hline &1&-5&\color{orangered}{1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 1 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&2&-34&9\\& & -7& 35& \color{blue}{-7} \\ \hline &1&-5&\color{blue}{1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-7&1&2&-34&\color{orangered}{ 9 }\\& & -7& 35& \color{orangered}{-7} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{1}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-5x+1 } $ with a remainder of $ \color{red}{ 2 } $.