Divide using synthetic division $$ ( -18x^{3}+14x^{2}-24x+12 ) \div ( 6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-18&14&-24&12\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-18}&\color{blue}{14}&\color{blue}{-24}&\color{orangered}{12} \end{array} $$The solution is:
$$ \frac{ -18x^{3}+14x^{2}-24x+12 }{ x } = \color{blue}{-18x^{2}+14x-24} ~+~ \frac{ \color{red}{ 12 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-18&14&-24&12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -18 }&14&-24&12\\& & & & \\ \hline &\color{orangered}{-18}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-18&14&-24&12\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-18}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 0 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}0&-18&\color{orangered}{ 14 }&-24&12\\& & \color{orangered}{0} & & \\ \hline &-18&\color{orangered}{14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 14 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-18&14&-24&12\\& & 0& \color{blue}{0} & \\ \hline &-18&\color{blue}{14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 0 } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}0&-18&14&\color{orangered}{ -24 }&12\\& & 0& \color{orangered}{0} & \\ \hline &-18&14&\color{orangered}{-24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-18&14&-24&12\\& & 0& 0& \color{blue}{0} \\ \hline &-18&14&\color{blue}{-24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 0 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrr}0&-18&14&-24&\color{orangered}{ 12 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-18}&\color{blue}{14}&\color{blue}{-24}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -18x^{2}+14x-24 } $ with a remainder of $ \color{red}{ 12 } $.