Divide using synthetic division $$ ( 2x^{4}-8x^{3}-27x^{2}+14x+24 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}6&2&-8&-27&14&24\\& & 12& 24& -18& \color{black}{-24} \\ \hline &\color{blue}{2}&\color{blue}{4}&\color{blue}{-3}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-8x^{3}-27x^{2}+14x+24 }{ x-6 } = \color{blue}{2x^{3}+4x^{2}-3x-4} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&2&-8&-27&14&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}6&\color{orangered}{ 2 }&-8&-27&14&24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 2 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&2&-8&-27&14&24\\& & \color{blue}{12} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 12 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}6&2&\color{orangered}{ -8 }&-27&14&24\\& & \color{orangered}{12} & & & \\ \hline &2&\color{orangered}{4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 4 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&2&-8&-27&14&24\\& & 12& \color{blue}{24} & & \\ \hline &2&\color{blue}{4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 24 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}6&2&-8&\color{orangered}{ -27 }&14&24\\& & 12& \color{orangered}{24} & & \\ \hline &2&4&\color{orangered}{-3}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&2&-8&-27&14&24\\& & 12& 24& \color{blue}{-18} & \\ \hline &2&4&\color{blue}{-3}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}6&2&-8&-27&\color{orangered}{ 14 }&24\\& & 12& 24& \color{orangered}{-18} & \\ \hline &2&4&-3&\color{orangered}{-4}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&2&-8&-27&14&24\\& & 12& 24& -18& \color{blue}{-24} \\ \hline &2&4&-3&\color{blue}{-4}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}6&2&-8&-27&14&\color{orangered}{ 24 }\\& & 12& 24& -18& \color{orangered}{-24} \\ \hline &\color{blue}{2}&\color{blue}{4}&\color{blue}{-3}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+4x^{2}-3x-4 } $ with a remainder of $ \color{red}{ 0 } $.