Divide using synthetic division $$ ( 24x^{4}+18x^{2}-30x ) \div ( 6x ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&24&0&18&-30&0\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{24}&\color{blue}{0}&\color{blue}{18}&\color{blue}{-30}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 24x^{4}+18x^{2}-30x }{ x } = \color{blue}{24x^{3}+18x-30} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&24&0&18&-30&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 24 }&0&18&-30&0\\& & & & & \\ \hline &\color{orangered}{24}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 24 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&24&0&18&-30&0\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{24}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&24&\color{orangered}{ 0 }&18&-30&0\\& & \color{orangered}{0} & & & \\ \hline &24&\color{orangered}{0}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&24&0&18&-30&0\\& & 0& \color{blue}{0} & & \\ \hline &24&\color{blue}{0}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 0 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrr}0&24&0&\color{orangered}{ 18 }&-30&0\\& & 0& \color{orangered}{0} & & \\ \hline &24&0&\color{orangered}{18}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&24&0&18&-30&0\\& & 0& 0& \color{blue}{0} & \\ \hline &24&0&\color{blue}{18}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 0 } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrrr}0&24&0&18&\color{orangered}{ -30 }&0\\& & 0& 0& \color{orangered}{0} & \\ \hline &24&0&18&\color{orangered}{-30}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&24&0&18&-30&0\\& & 0& 0& 0& \color{blue}{0} \\ \hline &24&0&18&\color{blue}{-30}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&24&0&18&-30&\color{orangered}{ 0 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{24}&\color{blue}{0}&\color{blue}{18}&\color{blue}{-30}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 24x^{3}+18x-30 } $ with a remainder of $ \color{red}{ 0 } $.