Divide using synthetic division $$ ( 8x^{3}+27 ) \div ( 2x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&8&0&0&27\\& & -24& 72& \color{black}{-216} \\ \hline &\color{blue}{8}&\color{blue}{-24}&\color{blue}{72}&\color{orangered}{-189} \end{array} $$The solution is:
$$ \frac{ 8x^{3}+27 }{ x+3 } = \color{blue}{8x^{2}-24x+72} \color{red}{~-~} \frac{ \color{red}{ 189 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&0&0&27\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 8 }&0&0&27\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 8 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&0&0&27\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}-3&8&\color{orangered}{ 0 }&0&27\\& & \color{orangered}{-24} & & \\ \hline &8&\color{orangered}{-24}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&0&0&27\\& & -24& \color{blue}{72} & \\ \hline &8&\color{blue}{-24}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 72 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrr}-3&8&0&\color{orangered}{ 0 }&27\\& & -24& \color{orangered}{72} & \\ \hline &8&-24&\color{orangered}{72}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 72 } = \color{blue}{ -216 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&0&0&27\\& & -24& 72& \color{blue}{-216} \\ \hline &8&-24&\color{blue}{72}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ \left( -216 \right) } = \color{orangered}{ -189 } $
$$ \begin{array}{c|rrrr}-3&8&0&0&\color{orangered}{ 27 }\\& & -24& 72& \color{orangered}{-216} \\ \hline &\color{blue}{8}&\color{blue}{-24}&\color{blue}{72}&\color{orangered}{-189} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-24x+72 } $ with a remainder of $ \color{red}{ -189 } $.