Divide using synthetic division $$ ( 12x^{3}-28x^{2}-6 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&12&-28&0&-6\\& & 36& 24& \color{black}{72} \\ \hline &\color{blue}{12}&\color{blue}{8}&\color{blue}{24}&\color{orangered}{66} \end{array} $$The solution is:
$$ \frac{ 12x^{3}-28x^{2}-6 }{ x-3 } = \color{blue}{12x^{2}+8x+24} ~+~ \frac{ \color{red}{ 66 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&-28&0&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 12 }&-28&0&-6\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 12 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&-28&0&-6\\& & \color{blue}{36} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 36 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}3&12&\color{orangered}{ -28 }&0&-6\\& & \color{orangered}{36} & & \\ \hline &12&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 8 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&-28&0&-6\\& & 36& \color{blue}{24} & \\ \hline &12&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 24 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}3&12&-28&\color{orangered}{ 0 }&-6\\& & 36& \color{orangered}{24} & \\ \hline &12&8&\color{orangered}{24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 24 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&-28&0&-6\\& & 36& 24& \color{blue}{72} \\ \hline &12&8&\color{blue}{24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 72 } = \color{orangered}{ 66 } $
$$ \begin{array}{c|rrrr}3&12&-28&0&\color{orangered}{ -6 }\\& & 36& 24& \color{orangered}{72} \\ \hline &\color{blue}{12}&\color{blue}{8}&\color{blue}{24}&\color{orangered}{66} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 12x^{2}+8x+24 } $ with a remainder of $ \color{red}{ 66 } $.