Divide using synthetic division $$ ( x^{3}+x^{2}-21x-27 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&1&1&-21&-27\\& & 6& 42& \color{black}{126} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{blue}{21}&\color{orangered}{99} \end{array} $$The solution is:
$$ \frac{ x^{3}+x^{2}-21x-27 }{ x-6 } = \color{blue}{x^{2}+7x+21} ~+~ \frac{ \color{red}{ 99 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&1&-21&-27\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 1 }&1&-21&-27\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&1&-21&-27\\& & \color{blue}{6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 6 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}6&1&\color{orangered}{ 1 }&-21&-27\\& & \color{orangered}{6} & & \\ \hline &1&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 7 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&1&-21&-27\\& & 6& \color{blue}{42} & \\ \hline &1&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 42 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrr}6&1&1&\color{orangered}{ -21 }&-27\\& & 6& \color{orangered}{42} & \\ \hline &1&7&\color{orangered}{21}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 21 } = \color{blue}{ 126 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&1&-21&-27\\& & 6& 42& \color{blue}{126} \\ \hline &1&7&\color{blue}{21}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 126 } = \color{orangered}{ 99 } $
$$ \begin{array}{c|rrrr}6&1&1&-21&\color{orangered}{ -27 }\\& & 6& 42& \color{orangered}{126} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{blue}{21}&\color{orangered}{99} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+7x+21 } $ with a remainder of $ \color{red}{ 99 } $.