Divide using synthetic division $$ ( -10x^{2}+15x ) \div ( -2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&-10&15&0\\& & 0& \color{black}{0} \\ \hline &\color{blue}{-10}&\color{blue}{15}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -10x^{2}+15x }{ x } = \color{blue}{-10x+15} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&-10&15&0\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ -10 }&15&0\\& & & \\ \hline &\color{orangered}{-10}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-10&15&0\\& & \color{blue}{0} & \\ \hline &\color{blue}{-10}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ 0 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrr}0&-10&\color{orangered}{ 15 }&0\\& & \color{orangered}{0} & \\ \hline &-10&\color{orangered}{15}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 15 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-10&15&0\\& & 0& \color{blue}{0} \\ \hline &-10&\color{blue}{15}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}0&-10&15&\color{orangered}{ 0 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{-10}&\color{blue}{15}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x+15 } $ with a remainder of $ \color{red}{ 0 } $.