Divide using synthetic division $$ ( 6x^{3}+47x^{2}+2x+84 ) \div ( 7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&6&47&2&84\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{6}&\color{blue}{47}&\color{blue}{2}&\color{orangered}{84} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+47x^{2}+2x+84 }{ x } = \color{blue}{6x^{2}+47x+2} ~+~ \frac{ \color{red}{ 84 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&6&47&2&84\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ 6 }&47&2&84\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 6 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&6&47&2&84\\& & \color{blue}{0} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 47 } + \color{orangered}{ 0 } = \color{orangered}{ 47 } $
$$ \begin{array}{c|rrrr}0&6&\color{orangered}{ 47 }&2&84\\& & \color{orangered}{0} & & \\ \hline &6&\color{orangered}{47}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 47 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&6&47&2&84\\& & 0& \color{blue}{0} & \\ \hline &6&\color{blue}{47}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}0&6&47&\color{orangered}{ 2 }&84\\& & 0& \color{orangered}{0} & \\ \hline &6&47&\color{orangered}{2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 2 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&6&47&2&84\\& & 0& 0& \color{blue}{0} \\ \hline &6&47&\color{blue}{2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 84 } + \color{orangered}{ 0 } = \color{orangered}{ 84 } $
$$ \begin{array}{c|rrrr}0&6&47&2&\color{orangered}{ 84 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{6}&\color{blue}{47}&\color{blue}{2}&\color{orangered}{84} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+47x+2 } $ with a remainder of $ \color{red}{ 84 } $.