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