Divide using synthetic division $$ ( 3x^{4}+5x^{3}-9x^{2}-10x+8 ) \div ( 1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&3&5&-9&-10&8\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{blue}{-9}&\color{blue}{-10}&\color{orangered}{8} \end{array} $$The solution is:
$$ \frac{ 3x^{4}+5x^{3}-9x^{2}-10x+8 }{ x } = \color{blue}{3x^{3}+5x^{2}-9x-10} ~+~ \frac{ \color{red}{ 8 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&5&-9&-10&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 3 }&5&-9&-10&8\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 3 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&5&-9&-10&8\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 0 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}0&3&\color{orangered}{ 5 }&-9&-10&8\\& & \color{orangered}{0} & & & \\ \hline &3&\color{orangered}{5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 5 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&5&-9&-10&8\\& & 0& \color{blue}{0} & & \\ \hline &3&\color{blue}{5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 0 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}0&3&5&\color{orangered}{ -9 }&-10&8\\& & 0& \color{orangered}{0} & & \\ \hline &3&5&\color{orangered}{-9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&5&-9&-10&8\\& & 0& 0& \color{blue}{0} & \\ \hline &3&5&\color{blue}{-9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 0 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}0&3&5&-9&\color{orangered}{ -10 }&8\\& & 0& 0& \color{orangered}{0} & \\ \hline &3&5&-9&\color{orangered}{-10}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&5&-9&-10&8\\& & 0& 0& 0& \color{blue}{0} \\ \hline &3&5&-9&\color{blue}{-10}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 0 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}0&3&5&-9&-10&\color{orangered}{ 8 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{blue}{-9}&\color{blue}{-10}&\color{orangered}{8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+5x^{2}-9x-10 } $ with a remainder of $ \color{red}{ 8 } $.