A rational expression is a quotient of two polynomials:

$$\frac{P(x)}{Q(x)},$$

where the denominator $Q(x)$ cannot be zero.

Domain and excluded values

Before simplifying a rational expression, it is important to identify the values that make the denominator zero. For example,

$$\frac{x^2-1}{x-1}$$

is undefined at $x=1$.

Factoring the numerator gives

$$\frac{(x-1)(x+1)}{x-1}=x+1,$$

but the simplification is valid only for $x\ne1$. The simplified expression has the same values as the original wherever the original is defined, but the excluded value does not disappear.

Arithmetic with rational expressions

Products and quotients can often be simplified by factoring before multiplying. Sums and differences require a common denominator, just as ordinary fractions do.

For example,

$$\frac1x+\frac1{x+1} =\frac{x+1+x}{x(x+1)} =\frac{2x+1}{x(x+1)},$$

with $x\ne0$ and $x\ne-1$.

Keeping track of excluded values is part of the expression, not an optional final check.