Divide using synthetic division $$ ( 27x^{3}+8 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&27&0&0&8\\& & -54& 108& \color{black}{-216} \\ \hline &\color{blue}{27}&\color{blue}{-54}&\color{blue}{108}&\color{orangered}{-208} \end{array} $$The solution is:
$$ \frac{ 27x^{3}+8 }{ x+2 } = \color{blue}{27x^{2}-54x+108} \color{red}{~-~} \frac{ \color{red}{ 208 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&27&0&0&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 27 }&0&0&8\\& & & & \\ \hline &\color{orangered}{27}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 27 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&27&0&0&8\\& & \color{blue}{-54} & & \\ \hline &\color{blue}{27}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -54 } $
$$ \begin{array}{c|rrrr}-2&27&\color{orangered}{ 0 }&0&8\\& & \color{orangered}{-54} & & \\ \hline &27&\color{orangered}{-54}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -54 \right) } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&27&0&0&8\\& & -54& \color{blue}{108} & \\ \hline &27&\color{blue}{-54}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 108 } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrrr}-2&27&0&\color{orangered}{ 0 }&8\\& & -54& \color{orangered}{108} & \\ \hline &27&-54&\color{orangered}{108}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 108 } = \color{blue}{ -216 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&27&0&0&8\\& & -54& 108& \color{blue}{-216} \\ \hline &27&-54&\color{blue}{108}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -216 \right) } = \color{orangered}{ -208 } $
$$ \begin{array}{c|rrrr}-2&27&0&0&\color{orangered}{ 8 }\\& & -54& 108& \color{orangered}{-216} \\ \hline &\color{blue}{27}&\color{blue}{-54}&\color{blue}{108}&\color{orangered}{-208} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 27x^{2}-54x+108 } $ with a remainder of $ \color{red}{ -208 } $.