Divide using synthetic division $$ ( x^{5}+4x^{4}+7x^{2}+27x-28 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&1&4&0&7&27&-28\\& & -4& 0& 0& -28& \color{black}{4} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{7}&\color{blue}{-1}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ x^{5}+4x^{4}+7x^{2}+27x-28 }{ x+4 } = \color{blue}{x^{4}+7x-1} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&4&0&7&27&-28\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 1 }&4&0&7&27&-28\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&4&0&7&27&-28\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-4&1&\color{orangered}{ 4 }&0&7&27&-28\\& & \color{orangered}{-4} & & & & \\ \hline &1&\color{orangered}{0}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&4&0&7&27&-28\\& & -4& \color{blue}{0} & & & \\ \hline &1&\color{blue}{0}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-4&1&4&\color{orangered}{ 0 }&7&27&-28\\& & -4& \color{orangered}{0} & & & \\ \hline &1&0&\color{orangered}{0}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&4&0&7&27&-28\\& & -4& 0& \color{blue}{0} & & \\ \hline &1&0&\color{blue}{0}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 0 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrr}-4&1&4&0&\color{orangered}{ 7 }&27&-28\\& & -4& 0& \color{orangered}{0} & & \\ \hline &1&0&0&\color{orangered}{7}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 7 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&4&0&7&27&-28\\& & -4& 0& 0& \color{blue}{-28} & \\ \hline &1&0&0&\color{blue}{7}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}-4&1&4&0&7&\color{orangered}{ 27 }&-28\\& & -4& 0& 0& \color{orangered}{-28} & \\ \hline &1&0&0&7&\color{orangered}{-1}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&4&0&7&27&-28\\& & -4& 0& 0& -28& \color{blue}{4} \\ \hline &1&0&0&7&\color{blue}{-1}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 4 } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrrr}-4&1&4&0&7&27&\color{orangered}{ -28 }\\& & -4& 0& 0& -28& \color{orangered}{4} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{7}&\color{blue}{-1}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+7x-1 } $ with a remainder of $ \color{red}{ -24 } $.