Divide using synthetic division $$ ( 6x^{3}-17x^{2}+6x+8 ) \div ( 3x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&6&-17&6&8\\& & 24& 28& \color{black}{136} \\ \hline &\color{blue}{6}&\color{blue}{7}&\color{blue}{34}&\color{orangered}{144} \end{array} $$The solution is:
$$ \frac{ 6x^{3}-17x^{2}+6x+8 }{ x-4 } = \color{blue}{6x^{2}+7x+34} ~+~ \frac{ \color{red}{ 144 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&-17&6&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 6 }&-17&6&8\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&-17&6&8\\& & \color{blue}{24} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 24 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}4&6&\color{orangered}{ -17 }&6&8\\& & \color{orangered}{24} & & \\ \hline &6&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 7 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&-17&6&8\\& & 24& \color{blue}{28} & \\ \hline &6&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 28 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrr}4&6&-17&\color{orangered}{ 6 }&8\\& & 24& \color{orangered}{28} & \\ \hline &6&7&\color{orangered}{34}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 34 } = \color{blue}{ 136 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&6&-17&6&8\\& & 24& 28& \color{blue}{136} \\ \hline &6&7&\color{blue}{34}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 136 } = \color{orangered}{ 144 } $
$$ \begin{array}{c|rrrr}4&6&-17&6&\color{orangered}{ 8 }\\& & 24& 28& \color{orangered}{136} \\ \hline &\color{blue}{6}&\color{blue}{7}&\color{blue}{34}&\color{orangered}{144} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+7x+34 } $ with a remainder of $ \color{red}{ 144 } $.