Divide using synthetic division $$ ( 3x^{3}+23x^{2}-9x+2 ) \div ( x+8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-8&3&23&-9&2\\& & -24& 8& \color{black}{8} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-1}&\color{orangered}{10} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+23x^{2}-9x+2 }{ x+8 } = \color{blue}{3x^{2}-x-1} ~+~ \frac{ \color{red}{ 10 } }{ x+8 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 8 = 0 $ ( $ x = \color{blue}{ -8 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&3&23&-9&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-8&\color{orangered}{ 3 }&23&-9&2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 3 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&3&23&-9&2\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-8&3&\color{orangered}{ 23 }&-9&2\\& & \color{orangered}{-24} & & \\ \hline &3&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&3&23&-9&2\\& & -24& \color{blue}{8} & \\ \hline &3&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 8 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-8&3&23&\color{orangered}{ -9 }&2\\& & -24& \color{orangered}{8} & \\ \hline &3&-1&\color{orangered}{-1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&3&23&-9&2\\& & -24& 8& \color{blue}{8} \\ \hline &3&-1&\color{blue}{-1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 8 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}-8&3&23&-9&\color{orangered}{ 2 }\\& & -24& 8& \color{orangered}{8} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-1}&\color{orangered}{10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-x-1 } $ with a remainder of $ \color{red}{ 10 } $.