Find remainder when $ 8x^{2}+27 $ is divided by $ 2x+3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-3&8&0&27\\& & -24& \color{black}{72} \\ \hline &\color{blue}{8}&\color{blue}{-24}&\color{orangered}{99} \end{array} $$The remainder when $ 8x^{2}+27 $ is divided by $ x+3 $ is $ \, \color{red}{ 99 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrr}\color{blue}{-3}&8&0&27\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-3&\color{orangered}{ 8 }&0&27\\& & & \\ \hline &\color{orangered}{8}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&8&0&27\\& & \color{blue}{-24} & \\ \hline &\color{blue}{8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrr}-3&8&\color{orangered}{ 0 }&27\\& & \color{orangered}{-24} & \\ \hline &8&\color{orangered}{-24}& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&8&0&27\\& & -24& \color{blue}{72} \\ \hline &8&\color{blue}{-24}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ 72 } = \color{orangered}{ 99 } $
$$ \begin{array}{c|rrr}-3&8&0&\color{orangered}{ 27 }\\& & -24& \color{orangered}{72} \\ \hline &\color{blue}{8}&\color{blue}{-24}&\color{orangered}{99} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 99 }\right) $.