Divide using synthetic division $$ ( -24x^{3}-33x^{2}+20x+3 ) \div ( -5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-24&-33&20&3\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-24}&\color{blue}{-33}&\color{blue}{20}&\color{orangered}{3} \end{array} $$The solution is:
$$ \frac{ -24x^{3}-33x^{2}+20x+3 }{ x } = \color{blue}{-24x^{2}-33x+20} ~+~ \frac{ \color{red}{ 3 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-24&-33&20&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -24 }&-33&20&3\\& & & & \\ \hline &\color{orangered}{-24}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-24&-33&20&3\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-24}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ 0 } = \color{orangered}{ -33 } $
$$ \begin{array}{c|rrrr}0&-24&\color{orangered}{ -33 }&20&3\\& & \color{orangered}{0} & & \\ \hline &-24&\color{orangered}{-33}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -33 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-24&-33&20&3\\& & 0& \color{blue}{0} & \\ \hline &-24&\color{blue}{-33}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ 0 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrr}0&-24&-33&\color{orangered}{ 20 }&3\\& & 0& \color{orangered}{0} & \\ \hline &-24&-33&\color{orangered}{20}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 20 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-24&-33&20&3\\& & 0& 0& \color{blue}{0} \\ \hline &-24&-33&\color{blue}{20}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}0&-24&-33&20&\color{orangered}{ 3 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-24}&\color{blue}{-33}&\color{blue}{20}&\color{orangered}{3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -24x^{2}-33x+20 } $ with a remainder of $ \color{red}{ 3 } $.