Manipulating Rational Expressions

A rational expression is undefined when its denominator equals $0$. Thus, we must find a denominator which equals $0$ if $x=5$ or $x=\text{-}7$. The simplest polynomial that gives $0$ when $x=5$ is $$5-x,$$ because $5-5=0$. By the same reasoning, we find $$7+x,$$ because the $7+(\text{-}7)=0$. For the denominator to be undefined for both $x=5$ and $x=\text{-}7$, the entire denominator must equal $0$ when substituting $x$ for these values. This is satisfied by the product $(5-x)(7+x).$ The numerator can be any polynomial, for example, choose the rational expression $$\frac{2x}{(5-x)(7+x)}.$$