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