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