Divide using synthetic division $$ ( 8x^{3}-14x^{2}+18x-9 ) \div ( -4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&8&-14&18&-9\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{8}&\color{blue}{-14}&\color{blue}{18}&\color{orangered}{-9} \end{array} $$The solution is:
$$ \frac{ 8x^{3}-14x^{2}+18x-9 }{ x } = \color{blue}{8x^{2}-14x+18} \color{red}{~-~} \frac{ \color{red}{ 9 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&8&-14&18&-9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ 8 }&-14&18&-9\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 8 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&8&-14&18&-9\\& & \color{blue}{0} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 0 } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrr}0&8&\color{orangered}{ -14 }&18&-9\\& & \color{orangered}{0} & & \\ \hline &8&\color{orangered}{-14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&8&-14&18&-9\\& & 0& \color{blue}{0} & \\ \hline &8&\color{blue}{-14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 0 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}0&8&-14&\color{orangered}{ 18 }&-9\\& & 0& \color{orangered}{0} & \\ \hline &8&-14&\color{orangered}{18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 18 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&8&-14&18&-9\\& & 0& 0& \color{blue}{0} \\ \hline &8&-14&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 0 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}0&8&-14&18&\color{orangered}{ -9 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{8}&\color{blue}{-14}&\color{blue}{18}&\color{orangered}{-9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-14x+18 } $ with a remainder of $ \color{red}{ -9 } $.