Divide using synthetic division $$ ( x^{3}-6x^{2}-32x+34 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&1&-6&-32&34\\& & -3& 27& \color{black}{15} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{-5}&\color{orangered}{49} \end{array} $$The solution is:
$$ \frac{ x^{3}-6x^{2}-32x+34 }{ x+3 } = \color{blue}{x^{2}-9x-5} ~+~ \frac{ \color{red}{ 49 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-6&-32&34\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 1 }&-6&-32&34\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-6&-32&34\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-3&1&\color{orangered}{ -6 }&-32&34\\& & \color{orangered}{-3} & & \\ \hline &1&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-6&-32&34\\& & -3& \color{blue}{27} & \\ \hline &1&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 27 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-3&1&-6&\color{orangered}{ -32 }&34\\& & -3& \color{orangered}{27} & \\ \hline &1&-9&\color{orangered}{-5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&-6&-32&34\\& & -3& 27& \color{blue}{15} \\ \hline &1&-9&\color{blue}{-5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ 15 } = \color{orangered}{ 49 } $
$$ \begin{array}{c|rrrr}-3&1&-6&-32&\color{orangered}{ 34 }\\& & -3& 27& \color{orangered}{15} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{-5}&\color{orangered}{49} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-9x-5 } $ with a remainder of $ \color{red}{ 49 } $.