Divide using synthetic division $$ ( 2x^{2}-16x+19 ) \div ( x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}7&2&-16&19\\& & 14& \color{black}{-14} \\ \hline &\color{blue}{2}&\color{blue}{-2}&\color{orangered}{5} \end{array} $$The solution is:
$$ \frac{ 2x^{2}-16x+19 }{ x-7 } = \color{blue}{2x-2} ~+~ \frac{ \color{red}{ 5 } }{ x-7 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{7}&2&-16&19\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}7&\color{orangered}{ 2 }&-16&19\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&2&-16&19\\& & \color{blue}{14} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 14 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrr}7&2&\color{orangered}{ -16 }&19\\& & \color{orangered}{14} & \\ \hline &2&\color{orangered}{-2}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&2&-16&19\\& & 14& \color{blue}{-14} \\ \hline &2&\color{blue}{-2}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}7&2&-16&\color{orangered}{ 19 }\\& & 14& \color{orangered}{-14} \\ \hline &\color{blue}{2}&\color{blue}{-2}&\color{orangered}{5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-2 } $ with a remainder of $ \color{red}{ 5 } $.