Divide using synthetic division $$ ( 7x^{2}+5x-200 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&7&5&-200\\& & 35& \color{black}{200} \\ \hline &\color{blue}{7}&\color{blue}{40}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 7x^{2}+5x-200 }{ x-5 } = \color{blue}{7x+40} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{5}&7&5&-200\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ 7 }&5&-200\\& & & \\ \hline &\color{orangered}{7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 7 } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&7&5&-200\\& & \color{blue}{35} & \\ \hline &\color{blue}{7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 35 } = \color{orangered}{ 40 } $
$$ \begin{array}{c|rrr}5&7&\color{orangered}{ 5 }&-200\\& & \color{orangered}{35} & \\ \hline &7&\color{orangered}{40}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 40 } = \color{blue}{ 200 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&7&5&-200\\& & 35& \color{blue}{200} \\ \hline &7&\color{blue}{40}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -200 } + \color{orangered}{ 200 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}5&7&5&\color{orangered}{ -200 }\\& & 35& \color{orangered}{200} \\ \hline &\color{blue}{7}&\color{blue}{40}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x+40 } $ with a remainder of $ \color{red}{ 0 } $.