Divide using synthetic division $$ ( 8x^{3}-66x^{2}+14x+8 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}8&8&-66&14&8\\& & 64& -16& \color{black}{-16} \\ \hline &\color{blue}{8}&\color{blue}{-2}&\color{blue}{-2}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ 8x^{3}-66x^{2}+14x+8 }{ x-8 } = \color{blue}{8x^{2}-2x-2} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x-8 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{8}&8&-66&14&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}8&\color{orangered}{ 8 }&-66&14&8\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 8 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&8&-66&14&8\\& & \color{blue}{64} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -66 } + \color{orangered}{ 64 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}8&8&\color{orangered}{ -66 }&14&8\\& & \color{orangered}{64} & & \\ \hline &8&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&8&-66&14&8\\& & 64& \color{blue}{-16} & \\ \hline &8&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}8&8&-66&\color{orangered}{ 14 }&8\\& & 64& \color{orangered}{-16} & \\ \hline &8&-2&\color{orangered}{-2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&8&-66&14&8\\& & 64& -16& \color{blue}{-16} \\ \hline &8&-2&\color{blue}{-2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}8&8&-66&14&\color{orangered}{ 8 }\\& & 64& -16& \color{orangered}{-16} \\ \hline &\color{blue}{8}&\color{blue}{-2}&\color{blue}{-2}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-2x-2 } $ with a remainder of $ \color{red}{ -8 } $.