Divide using synthetic division $$ ( 25x^{2}-49 ) \div ( 5x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}7&25&0&-49\\& & 175& \color{black}{1225} \\ \hline &\color{blue}{25}&\color{blue}{175}&\color{orangered}{1176} \end{array} $$The solution is:
$$ \frac{ 25x^{2}-49 }{ x-7 } = \color{blue}{25x+175} ~+~ \frac{ \color{red}{ 1176 } }{ x-7 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{7}&25&0&-49\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}7&\color{orangered}{ 25 }&0&-49\\& & & \\ \hline &\color{orangered}{25}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 25 } = \color{blue}{ 175 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&25&0&-49\\& & \color{blue}{175} & \\ \hline &\color{blue}{25}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 175 } = \color{orangered}{ 175 } $
$$ \begin{array}{c|rrr}7&25&\color{orangered}{ 0 }&-49\\& & \color{orangered}{175} & \\ \hline &25&\color{orangered}{175}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 175 } = \color{blue}{ 1225 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&25&0&-49\\& & 175& \color{blue}{1225} \\ \hline &25&\color{blue}{175}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -49 } + \color{orangered}{ 1225 } = \color{orangered}{ 1176 } $
$$ \begin{array}{c|rrr}7&25&0&\color{orangered}{ -49 }\\& & 175& \color{orangered}{1225} \\ \hline &\color{blue}{25}&\color{blue}{175}&\color{orangered}{1176} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 25x+175 } $ with a remainder of $ \color{red}{ 1176 } $.