Divide using synthetic division $$ ( 3x^{3}+9x^{2}-5x-7 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&3&9&-5&-7\\& & -9& 0& \color{black}{15} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{-5}&\color{orangered}{8} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+9x^{2}-5x-7 }{ x+3 } = \color{blue}{3x^{2}-5} ~+~ \frac{ \color{red}{ 8 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&9&-5&-7\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 3 }&9&-5&-7\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&9&-5&-7\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-3&3&\color{orangered}{ 9 }&-5&-7\\& & \color{orangered}{-9} & & \\ \hline &3&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&9&-5&-7\\& & -9& \color{blue}{0} & \\ \hline &3&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 0 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-3&3&9&\color{orangered}{ -5 }&-7\\& & -9& \color{orangered}{0} & \\ \hline &3&0&\color{orangered}{-5}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&9&-5&-7\\& & -9& 0& \color{blue}{15} \\ \hline &3&0&\color{blue}{-5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 15 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}-3&3&9&-5&\color{orangered}{ -7 }\\& & -9& 0& \color{orangered}{15} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{-5}&\color{orangered}{8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-5 } $ with a remainder of $ \color{red}{ 8 } $.