Divide using synthetic division $$ ( x^{4}+4x^{3}-7x^{2}-34x-24 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&1&4&-7&-34&-24\\& & -4& 0& 28& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{-7}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}+4x^{3}-7x^{2}-34x-24 }{ x+4 } = \color{blue}{x^{3}-7x-6} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&4&-7&-34&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 1 }&4&-7&-34&-24\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&4&-7&-34&-24\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-4&1&\color{orangered}{ 4 }&-7&-34&-24\\& & \color{orangered}{-4} & & & \\ \hline &1&\color{orangered}{0}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&4&-7&-34&-24\\& & -4& \color{blue}{0} & & \\ \hline &1&\color{blue}{0}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 0 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}-4&1&4&\color{orangered}{ -7 }&-34&-24\\& & -4& \color{orangered}{0} & & \\ \hline &1&0&\color{orangered}{-7}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&4&-7&-34&-24\\& & -4& 0& \color{blue}{28} & \\ \hline &1&0&\color{blue}{-7}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 28 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-4&1&4&-7&\color{orangered}{ -34 }&-24\\& & -4& 0& \color{orangered}{28} & \\ \hline &1&0&-7&\color{orangered}{-6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&4&-7&-34&-24\\& & -4& 0& 28& \color{blue}{24} \\ \hline &1&0&-7&\color{blue}{-6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-4&1&4&-7&-34&\color{orangered}{ -24 }\\& & -4& 0& 28& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{-7}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-7x-6 } $ with a remainder of $ \color{red}{ 0 } $.