Divide using synthetic division $$ ( 2x^{2}+7x-39 ) \div ( 2x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}7&2&7&-39\\& & 14& \color{black}{147} \\ \hline &\color{blue}{2}&\color{blue}{21}&\color{orangered}{108} \end{array} $$The solution is:
$$ \frac{ 2x^{2}+7x-39 }{ x-7 } = \color{blue}{2x+21} ~+~ \frac{ \color{red}{ 108 } }{ x-7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{7}&2&7&-39\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}7&\color{orangered}{ 2 }&7&-39\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 2 } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&2&7&-39\\& & \color{blue}{14} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 14 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrr}7&2&\color{orangered}{ 7 }&-39\\& & \color{orangered}{14} & \\ \hline &2&\color{orangered}{21}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 21 } = \color{blue}{ 147 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&2&7&-39\\& & 14& \color{blue}{147} \\ \hline &2&\color{blue}{21}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -39 } + \color{orangered}{ 147 } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrr}7&2&7&\color{orangered}{ -39 }\\& & 14& \color{orangered}{147} \\ \hline &\color{blue}{2}&\color{blue}{21}&\color{orangered}{108} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x+21 } $ with a remainder of $ \color{red}{ 108 } $.