Divide using synthetic division $$ ( 2x^{3}+14x^{2}-32x+36 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&2&14&-32&36\\& & -18& 36& \color{black}{-36} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+14x^{2}-32x+36 }{ x+9 } = \color{blue}{2x^{2}-4x+4} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&2&14&-32&36\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 2 }&14&-32&36\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 2 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&2&14&-32&36\\& & \color{blue}{-18} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-9&2&\color{orangered}{ 14 }&-32&36\\& & \color{orangered}{-18} & & \\ \hline &2&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&2&14&-32&36\\& & -18& \color{blue}{36} & \\ \hline &2&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 36 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-9&2&14&\color{orangered}{ -32 }&36\\& & -18& \color{orangered}{36} & \\ \hline &2&-4&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 4 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&2&14&-32&36\\& & -18& 36& \color{blue}{-36} \\ \hline &2&-4&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-9&2&14&-32&\color{orangered}{ 36 }\\& & -18& 36& \color{orangered}{-36} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-4x+4 } $ with a remainder of $ \color{red}{ 0 } $.