Divide using synthetic division $$ ( 2x^{3}-13x-4 ) \div ( x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&2&0&-13&-4\\& & 14& 98& \color{black}{595} \\ \hline &\color{blue}{2}&\color{blue}{14}&\color{blue}{85}&\color{orangered}{591} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-13x-4 }{ x-7 } = \color{blue}{2x^{2}+14x+85} ~+~ \frac{ \color{red}{ 591 } }{ x-7 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&0&-13&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 2 }&0&-13&-4\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 2 } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&0&-13&-4\\& & \color{blue}{14} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 14 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}7&2&\color{orangered}{ 0 }&-13&-4\\& & \color{orangered}{14} & & \\ \hline &2&\color{orangered}{14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 14 } = \color{blue}{ 98 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&0&-13&-4\\& & 14& \color{blue}{98} & \\ \hline &2&\color{blue}{14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 98 } = \color{orangered}{ 85 } $
$$ \begin{array}{c|rrrr}7&2&0&\color{orangered}{ -13 }&-4\\& & 14& \color{orangered}{98} & \\ \hline &2&14&\color{orangered}{85}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 85 } = \color{blue}{ 595 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&2&0&-13&-4\\& & 14& 98& \color{blue}{595} \\ \hline &2&14&\color{blue}{85}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 595 } = \color{orangered}{ 591 } $
$$ \begin{array}{c|rrrr}7&2&0&-13&\color{orangered}{ -4 }\\& & 14& 98& \color{orangered}{595} \\ \hline &\color{blue}{2}&\color{blue}{14}&\color{blue}{85}&\color{orangered}{591} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+14x+85 } $ with a remainder of $ \color{red}{ 591 } $.