Divide using synthetic division $$ ( 2x^{2}-13x+10 ) \div ( 5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&2&-13&10\\& & 0& \color{black}{0} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{10} \end{array} $$The solution is:
$$ \frac{ 2x^{2}-13x+10 }{ x } = \color{blue}{2x-13} ~+~ \frac{ \color{red}{ 10 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&2&-13&10\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ 2 }&-13&10\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 2 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&2&-13&10\\& & \color{blue}{0} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 0 } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrr}0&2&\color{orangered}{ -13 }&10\\& & \color{orangered}{0} & \\ \hline &2&\color{orangered}{-13}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&2&-13&10\\& & 0& \color{blue}{0} \\ \hline &2&\color{blue}{-13}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 0 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrr}0&2&-13&\color{orangered}{ 10 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-13 } $ with a remainder of $ \color{red}{ 10 } $.