Divide using synthetic division $$ ( 2x^{3}+x^{2}-18x ) \div ( 2x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&2&1&-18&0\\& & -2& 1& \color{black}{17} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-17}&\color{orangered}{17} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+x^{2}-18x }{ x+1 } = \color{blue}{2x^{2}-x-17} ~+~ \frac{ \color{red}{ 17 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&1&-18&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 2 }&1&-18&0\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&1&-18&0\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-1&2&\color{orangered}{ 1 }&-18&0\\& & \color{orangered}{-2} & & \\ \hline &2&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&1&-18&0\\& & -2& \color{blue}{1} & \\ \hline &2&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 1 } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}-1&2&1&\color{orangered}{ -18 }&0\\& & -2& \color{orangered}{1} & \\ \hline &2&-1&\color{orangered}{-17}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 17 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&1&-18&0\\& & -2& 1& \color{blue}{17} \\ \hline &2&-1&\color{blue}{-17}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 17 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrr}-1&2&1&-18&\color{orangered}{ 0 }\\& & -2& 1& \color{orangered}{17} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-17}&\color{orangered}{17} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-x-17 } $ with a remainder of $ \color{red}{ 17 } $.