Divide using synthetic division $$ ( 2x^{4}-4x^{2}-27x-18 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&2&0&-4&-27&-18\\& & -6& 18& -42& \color{black}{207} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{14}&\color{blue}{-69}&\color{orangered}{189} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-4x^{2}-27x-18 }{ x+3 } = \color{blue}{2x^{3}-6x^{2}+14x-69} ~+~ \frac{ \color{red}{ 189 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&-4&-27&-18\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 2 }&0&-4&-27&-18\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&-4&-27&-18\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-3&2&\color{orangered}{ 0 }&-4&-27&-18\\& & \color{orangered}{-6} & & & \\ \hline &2&\color{orangered}{-6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&-4&-27&-18\\& & -6& \color{blue}{18} & & \\ \hline &2&\color{blue}{-6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 18 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrr}-3&2&0&\color{orangered}{ -4 }&-27&-18\\& & -6& \color{orangered}{18} & & \\ \hline &2&-6&\color{orangered}{14}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 14 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&-4&-27&-18\\& & -6& 18& \color{blue}{-42} & \\ \hline &2&-6&\color{blue}{14}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -69 } $
$$ \begin{array}{c|rrrrr}-3&2&0&-4&\color{orangered}{ -27 }&-18\\& & -6& 18& \color{orangered}{-42} & \\ \hline &2&-6&14&\color{orangered}{-69}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -69 \right) } = \color{blue}{ 207 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&-4&-27&-18\\& & -6& 18& -42& \color{blue}{207} \\ \hline &2&-6&14&\color{blue}{-69}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 207 } = \color{orangered}{ 189 } $
$$ \begin{array}{c|rrrrr}-3&2&0&-4&-27&\color{orangered}{ -18 }\\& & -6& 18& -42& \color{orangered}{207} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{14}&\color{blue}{-69}&\color{orangered}{189} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-6x^{2}+14x-69 } $ with a remainder of $ \color{red}{ 189 } $.