Divide using synthetic division $$ ( x^{3}+7x^{2}-25x-175 ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&1&7&-25&-175\\& & -7& 0& \color{black}{175} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{-25}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+7x^{2}-25x-175 }{ x+7 } = \color{blue}{x^{2}-25} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&7&-25&-175\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ 1 }&7&-25&-175\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&7&-25&-175\\& & \color{blue}{-7} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-7&1&\color{orangered}{ 7 }&-25&-175\\& & \color{orangered}{-7} & & \\ \hline &1&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&7&-25&-175\\& & -7& \color{blue}{0} & \\ \hline &1&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 0 } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrr}-7&1&7&\color{orangered}{ -25 }&-175\\& & -7& \color{orangered}{0} & \\ \hline &1&0&\color{orangered}{-25}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -25 \right) } = \color{blue}{ 175 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&7&-25&-175\\& & -7& 0& \color{blue}{175} \\ \hline &1&0&\color{blue}{-25}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -175 } + \color{orangered}{ 175 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-7&1&7&-25&\color{orangered}{ -175 }\\& & -7& 0& \color{orangered}{175} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{-25}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-25 } $ with a remainder of $ \color{red}{ 0 } $.