Divide using synthetic division $$ ( 6x^{4}+49x^{3}+109x^{2}+16x-66 ) \div ( 6x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-7&6&49&109&16&-66\\& & -42& -49& -420& \color{black}{2828} \\ \hline &\color{blue}{6}&\color{blue}{7}&\color{blue}{60}&\color{blue}{-404}&\color{orangered}{2762} \end{array} $$The solution is:
$$ \frac{ 6x^{4}+49x^{3}+109x^{2}+16x-66 }{ x+7 } = \color{blue}{6x^{3}+7x^{2}+60x-404} ~+~ \frac{ \color{red}{ 2762 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&6&49&109&16&-66\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-7&\color{orangered}{ 6 }&49&109&16&-66\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 6 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&6&49&109&16&-66\\& & \color{blue}{-42} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 49 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}-7&6&\color{orangered}{ 49 }&109&16&-66\\& & \color{orangered}{-42} & & & \\ \hline &6&\color{orangered}{7}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 7 } = \color{blue}{ -49 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&6&49&109&16&-66\\& & -42& \color{blue}{-49} & & \\ \hline &6&\color{blue}{7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 109 } + \color{orangered}{ \left( -49 \right) } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrrrr}-7&6&49&\color{orangered}{ 109 }&16&-66\\& & -42& \color{orangered}{-49} & & \\ \hline &6&7&\color{orangered}{60}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 60 } = \color{blue}{ -420 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&6&49&109&16&-66\\& & -42& -49& \color{blue}{-420} & \\ \hline &6&7&\color{blue}{60}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -420 \right) } = \color{orangered}{ -404 } $
$$ \begin{array}{c|rrrrr}-7&6&49&109&\color{orangered}{ 16 }&-66\\& & -42& -49& \color{orangered}{-420} & \\ \hline &6&7&60&\color{orangered}{-404}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -404 \right) } = \color{blue}{ 2828 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&6&49&109&16&-66\\& & -42& -49& -420& \color{blue}{2828} \\ \hline &6&7&60&\color{blue}{-404}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -66 } + \color{orangered}{ 2828 } = \color{orangered}{ 2762 } $
$$ \begin{array}{c|rrrrr}-7&6&49&109&16&\color{orangered}{ -66 }\\& & -42& -49& -420& \color{orangered}{2828} \\ \hline &\color{blue}{6}&\color{blue}{7}&\color{blue}{60}&\color{blue}{-404}&\color{orangered}{2762} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}+7x^{2}+60x-404 } $ with a remainder of $ \color{red}{ 2762 } $.