Divide using synthetic division $$ ( 2x^{3}-9x^{2}+11x-12 ) \div ( 2x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&2&-9&11&-12\\& & 14& 35& \color{black}{322} \\ \hline &\color{blue}{2}&\color{blue}{5}&\color{blue}{46}&\color{orangered}{310} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-9x^{2}+11x-12 }{ x-7 } = \color{blue}{2x^{2}+5x+46} ~+~ \frac{ \color{red}{ 310 } }{ x-7 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&-9&11&-12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 2 }&-9&11&-12\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&-9&11&-12\\& & \color{blue}{14} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 14 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}7&2&\color{orangered}{ -9 }&11&-12\\& & \color{orangered}{14} & & \\ \hline &2&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 5 } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&-9&11&-12\\& & 14& \color{blue}{35} & \\ \hline &2&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 35 } = \color{orangered}{ 46 } $
$$ \begin{array}{c|rrrr}7&2&-9&\color{orangered}{ 11 }&-12\\& & 14& \color{orangered}{35} & \\ \hline &2&5&\color{orangered}{46}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 46 } = \color{blue}{ 322 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&-9&11&-12\\& & 14& 35& \color{blue}{322} \\ \hline &2&5&\color{blue}{46}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 322 } = \color{orangered}{ 310 } $
$$ \begin{array}{c|rrrr}7&2&-9&11&\color{orangered}{ -12 }\\& & 14& 35& \color{orangered}{322} \\ \hline &\color{blue}{2}&\color{blue}{5}&\color{blue}{46}&\color{orangered}{310} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+5x+46 } $ with a remainder of $ \color{red}{ 310 } $.