Divide using synthetic division $$ ( 3x^{2}-54x+42 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}6&3&-54&42\\& & 18& \color{black}{-216} \\ \hline &\color{blue}{3}&\color{blue}{-36}&\color{orangered}{-174} \end{array} $$The solution is:
$$ \frac{ 3x^{2}-54x+42 }{ x-6 } = \color{blue}{3x-36} \color{red}{~-~} \frac{ \color{red}{ 174 } }{ x-6 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{6}&3&-54&42\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}6&\color{orangered}{ 3 }&-54&42\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 3 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&3&-54&42\\& & \color{blue}{18} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 18 } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrr}6&3&\color{orangered}{ -54 }&42\\& & \color{orangered}{18} & \\ \hline &3&\color{orangered}{-36}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -36 \right) } = \color{blue}{ -216 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&3&-54&42\\& & 18& \color{blue}{-216} \\ \hline &3&\color{blue}{-36}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 42 } + \color{orangered}{ \left( -216 \right) } = \color{orangered}{ -174 } $
$$ \begin{array}{c|rrr}6&3&-54&\color{orangered}{ 42 }\\& & 18& \color{orangered}{-216} \\ \hline &\color{blue}{3}&\color{blue}{-36}&\color{orangered}{-174} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-36 } $ with a remainder of $ \color{red}{ -174 } $.