Divide using synthetic division $$ ( 3x^{3}+9x^{2}-x-14 ) \div ( 2x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&3&9&-1&-14\\& & 3& 12& \color{black}{11} \\ \hline &\color{blue}{3}&\color{blue}{12}&\color{blue}{11}&\color{orangered}{-3} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+9x^{2}-x-14 }{ x-1 } = \color{blue}{3x^{2}+12x+11} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&9&-1&-14\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 3 }&9&-1&-14\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 3 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&9&-1&-14\\& & \color{blue}{3} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 3 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrr}1&3&\color{orangered}{ 9 }&-1&-14\\& & \color{orangered}{3} & & \\ \hline &3&\color{orangered}{12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 12 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&9&-1&-14\\& & 3& \color{blue}{12} & \\ \hline &3&\color{blue}{12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 12 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}1&3&9&\color{orangered}{ -1 }&-14\\& & 3& \color{orangered}{12} & \\ \hline &3&12&\color{orangered}{11}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 11 } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&3&9&-1&-14\\& & 3& 12& \color{blue}{11} \\ \hline &3&12&\color{blue}{11}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 11 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}1&3&9&-1&\color{orangered}{ -14 }\\& & 3& 12& \color{orangered}{11} \\ \hline &\color{blue}{3}&\color{blue}{12}&\color{blue}{11}&\color{orangered}{-3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+12x+11 } $ with a remainder of $ \color{red}{ -3 } $.