Divide using synthetic division $$ ( -3x^{3}+13x+2 ) \div ( 5x ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-3&0&13&2\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-3}&\color{blue}{0}&\color{blue}{13}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ -3x^{3}+13x+2 }{ x } = \color{blue}{-3x^{2}+13} ~+~ \frac{ \color{red}{ 2 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-3&0&13&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -3 }&0&13&2\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-3&0&13&2\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}0&-3&\color{orangered}{ 0 }&13&2\\& & \color{orangered}{0} & & \\ \hline &-3&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-3&0&13&2\\& & 0& \color{blue}{0} & \\ \hline &-3&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ 0 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrr}0&-3&0&\color{orangered}{ 13 }&2\\& & 0& \color{orangered}{0} & \\ \hline &-3&0&\color{orangered}{13}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 13 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-3&0&13&2\\& & 0& 0& \color{blue}{0} \\ \hline &-3&0&\color{blue}{13}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}0&-3&0&13&\color{orangered}{ 2 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-3}&\color{blue}{0}&\color{blue}{13}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{2}+13 } $ with a remainder of $ \color{red}{ 2 } $.