Divide using synthetic division $$ ( x^{3}+6x^{2}-2x+225 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&1&6&-2&225\\& & -9& 27& \color{black}{-225} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{25}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+6x^{2}-2x+225 }{ x+9 } = \color{blue}{x^{2}-3x+25} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&6&-2&225\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 1 }&6&-2&225\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&6&-2&225\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-9&1&\color{orangered}{ 6 }&-2&225\\& & \color{orangered}{-9} & & \\ \hline &1&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&6&-2&225\\& & -9& \color{blue}{27} & \\ \hline &1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 27 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrr}-9&1&6&\color{orangered}{ -2 }&225\\& & -9& \color{orangered}{27} & \\ \hline &1&-3&\color{orangered}{25}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 25 } = \color{blue}{ -225 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&6&-2&225\\& & -9& 27& \color{blue}{-225} \\ \hline &1&-3&\color{blue}{25}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 225 } + \color{orangered}{ \left( -225 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-9&1&6&-2&\color{orangered}{ 225 }\\& & -9& 27& \color{orangered}{-225} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{25}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-3x+25 } $ with a remainder of $ \color{red}{ 0 } $.