Divide using synthetic division $$ ( -5x^{3}+39x^{2}-86x ) \div ( -5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-5&39&-86&0\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-5}&\color{blue}{39}&\color{blue}{-86}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -5x^{3}+39x^{2}-86x }{ x } = \color{blue}{-5x^{2}+39x-86} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&39&-86&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -5 }&39&-86&0\\& & & & \\ \hline &\color{orangered}{-5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&39&-86&0\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 39 } + \color{orangered}{ 0 } = \color{orangered}{ 39 } $
$$ \begin{array}{c|rrrr}0&-5&\color{orangered}{ 39 }&-86&0\\& & \color{orangered}{0} & & \\ \hline &-5&\color{orangered}{39}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 39 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&39&-86&0\\& & 0& \color{blue}{0} & \\ \hline &-5&\color{blue}{39}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -86 } + \color{orangered}{ 0 } = \color{orangered}{ -86 } $
$$ \begin{array}{c|rrrr}0&-5&39&\color{orangered}{ -86 }&0\\& & 0& \color{orangered}{0} & \\ \hline &-5&39&\color{orangered}{-86}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -86 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&39&-86&0\\& & 0& 0& \color{blue}{0} \\ \hline &-5&39&\color{blue}{-86}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}0&-5&39&-86&\color{orangered}{ 0 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-5}&\color{blue}{39}&\color{blue}{-86}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{2}+39x-86 } $ with a remainder of $ \color{red}{ 0 } $.