Divide using synthetic division $$ ( 28x^{4}-30x^{3}+6x^{2}-12x-28 ) \div ( 4x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}6&28&-30&6&-12&-28\\& & 168& 828& 5004& \color{black}{29952} \\ \hline &\color{blue}{28}&\color{blue}{138}&\color{blue}{834}&\color{blue}{4992}&\color{orangered}{29924} \end{array} $$The solution is:
$$ \frac{ 28x^{4}-30x^{3}+6x^{2}-12x-28 }{ x-6 } = \color{blue}{28x^{3}+138x^{2}+834x+4992} ~+~ \frac{ \color{red}{ 29924 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&28&-30&6&-12&-28\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}6&\color{orangered}{ 28 }&-30&6&-12&-28\\& & & & & \\ \hline &\color{orangered}{28}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 28 } = \color{blue}{ 168 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&28&-30&6&-12&-28\\& & \color{blue}{168} & & & \\ \hline &\color{blue}{28}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 168 } = \color{orangered}{ 138 } $
$$ \begin{array}{c|rrrrr}6&28&\color{orangered}{ -30 }&6&-12&-28\\& & \color{orangered}{168} & & & \\ \hline &28&\color{orangered}{138}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 138 } = \color{blue}{ 828 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&28&-30&6&-12&-28\\& & 168& \color{blue}{828} & & \\ \hline &28&\color{blue}{138}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 828 } = \color{orangered}{ 834 } $
$$ \begin{array}{c|rrrrr}6&28&-30&\color{orangered}{ 6 }&-12&-28\\& & 168& \color{orangered}{828} & & \\ \hline &28&138&\color{orangered}{834}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 834 } = \color{blue}{ 5004 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&28&-30&6&-12&-28\\& & 168& 828& \color{blue}{5004} & \\ \hline &28&138&\color{blue}{834}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 5004 } = \color{orangered}{ 4992 } $
$$ \begin{array}{c|rrrrr}6&28&-30&6&\color{orangered}{ -12 }&-28\\& & 168& 828& \color{orangered}{5004} & \\ \hline &28&138&834&\color{orangered}{4992}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 4992 } = \color{blue}{ 29952 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&28&-30&6&-12&-28\\& & 168& 828& 5004& \color{blue}{29952} \\ \hline &28&138&834&\color{blue}{4992}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 29952 } = \color{orangered}{ 29924 } $
$$ \begin{array}{c|rrrrr}6&28&-30&6&-12&\color{orangered}{ -28 }\\& & 168& 828& 5004& \color{orangered}{29952} \\ \hline &\color{blue}{28}&\color{blue}{138}&\color{blue}{834}&\color{blue}{4992}&\color{orangered}{29924} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 28x^{3}+138x^{2}+834x+4992 } $ with a remainder of $ \color{red}{ 29924 } $.