Divide using synthetic division $$ ( 6x^{3}+2x+131 ) \div ( x+8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-8&6&0&2&131\\& & -48& 384& \color{black}{-3088} \\ \hline &\color{blue}{6}&\color{blue}{-48}&\color{blue}{386}&\color{orangered}{-2957} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+2x+131 }{ x+8 } = \color{blue}{6x^{2}-48x+386} \color{red}{~-~} \frac{ \color{red}{ 2957 } }{ x+8 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&6&0&2&131\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-8&\color{orangered}{ 6 }&0&2&131\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 6 } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&6&0&2&131\\& & \color{blue}{-48} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrr}-8&6&\color{orangered}{ 0 }&2&131\\& & \color{orangered}{-48} & & \\ \hline &6&\color{orangered}{-48}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -48 \right) } = \color{blue}{ 384 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&6&0&2&131\\& & -48& \color{blue}{384} & \\ \hline &6&\color{blue}{-48}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 384 } = \color{orangered}{ 386 } $
$$ \begin{array}{c|rrrr}-8&6&0&\color{orangered}{ 2 }&131\\& & -48& \color{orangered}{384} & \\ \hline &6&-48&\color{orangered}{386}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 386 } = \color{blue}{ -3088 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&6&0&2&131\\& & -48& 384& \color{blue}{-3088} \\ \hline &6&-48&\color{blue}{386}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 131 } + \color{orangered}{ \left( -3088 \right) } = \color{orangered}{ -2957 } $
$$ \begin{array}{c|rrrr}-8&6&0&2&\color{orangered}{ 131 }\\& & -48& 384& \color{orangered}{-3088} \\ \hline &\color{blue}{6}&\color{blue}{-48}&\color{blue}{386}&\color{orangered}{-2957} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}-48x+386 } $ with a remainder of $ \color{red}{ -2957 } $.