Divide using synthetic division $$ ( x^{3}+8x^{2}-12x-27 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&1&8&-12&-27\\& & -9& 9& \color{black}{27} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+8x^{2}-12x-27 }{ x+9 } = \color{blue}{x^{2}-x-3} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&8&-12&-27\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 1 }&8&-12&-27\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 1 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&8&-12&-27\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-9&1&\color{orangered}{ 8 }&-12&-27\\& & \color{orangered}{-9} & & \\ \hline &1&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&8&-12&-27\\& & -9& \color{blue}{9} & \\ \hline &1&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 9 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-9&1&8&\color{orangered}{ -12 }&-27\\& & -9& \color{orangered}{9} & \\ \hline &1&-1&\color{orangered}{-3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&8&-12&-27\\& & -9& 9& \color{blue}{27} \\ \hline &1&-1&\color{blue}{-3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 27 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-9&1&8&-12&\color{orangered}{ -27 }\\& & -9& 9& \color{orangered}{27} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-x-3 } $ with a remainder of $ \color{red}{ 0 } $.