Divide using synthetic division $$ ( 32x^{3}-16x^{2}-18x+9 ) \div ( 2x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&32&-16&-18&9\\& & -32& 48& \color{black}{-30} \\ \hline &\color{blue}{32}&\color{blue}{-48}&\color{blue}{30}&\color{orangered}{-21} \end{array} $$The solution is:
$$ \frac{ 32x^{3}-16x^{2}-18x+9 }{ x+1 } = \color{blue}{32x^{2}-48x+30} \color{red}{~-~} \frac{ \color{red}{ 21 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&32&-16&-18&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 32 }&-16&-18&9\\& & & & \\ \hline &\color{orangered}{32}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 32 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&32&-16&-18&9\\& & \color{blue}{-32} & & \\ \hline &\color{blue}{32}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrr}-1&32&\color{orangered}{ -16 }&-18&9\\& & \color{orangered}{-32} & & \\ \hline &32&\color{orangered}{-48}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -48 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&32&-16&-18&9\\& & -32& \color{blue}{48} & \\ \hline &32&\color{blue}{-48}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 48 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrr}-1&32&-16&\color{orangered}{ -18 }&9\\& & -32& \color{orangered}{48} & \\ \hline &32&-48&\color{orangered}{30}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 30 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&32&-16&-18&9\\& & -32& 48& \color{blue}{-30} \\ \hline &32&-48&\color{blue}{30}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrr}-1&32&-16&-18&\color{orangered}{ 9 }\\& & -32& 48& \color{orangered}{-30} \\ \hline &\color{blue}{32}&\color{blue}{-48}&\color{blue}{30}&\color{orangered}{-21} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 32x^{2}-48x+30 } $ with a remainder of $ \color{red}{ -21 } $.