Divide using synthetic division $$ ( 2x^{3}-13x^{2}-42x-27 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}8&2&-13&-42&-27\\& & 16& 24& \color{black}{-144} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-18}&\color{orangered}{-171} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-13x^{2}-42x-27 }{ x-8 } = \color{blue}{2x^{2}+3x-18} \color{red}{~-~} \frac{ \color{red}{ 171 } }{ x-8 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{8}&2&-13&-42&-27\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}8&\color{orangered}{ 2 }&-13&-42&-27\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 2 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&2&-13&-42&-27\\& & \color{blue}{16} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 16 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}8&2&\color{orangered}{ -13 }&-42&-27\\& & \color{orangered}{16} & & \\ \hline &2&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 3 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&2&-13&-42&-27\\& & 16& \color{blue}{24} & \\ \hline &2&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -42 } + \color{orangered}{ 24 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrr}8&2&-13&\color{orangered}{ -42 }&-27\\& & 16& \color{orangered}{24} & \\ \hline &2&3&\color{orangered}{-18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ -144 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&2&-13&-42&-27\\& & 16& 24& \color{blue}{-144} \\ \hline &2&3&\color{blue}{-18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ \left( -144 \right) } = \color{orangered}{ -171 } $
$$ \begin{array}{c|rrrr}8&2&-13&-42&\color{orangered}{ -27 }\\& & 16& 24& \color{orangered}{-144} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-18}&\color{orangered}{-171} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+3x-18 } $ with a remainder of $ \color{red}{ -171 } $.