What a Rational Expression Is — and Where It Breaks

A fraction built from polynomials

A rational expression is a fraction whose numerator and denominator are both polynomials. Things like (x+2)/(x−3), or (x^2−1)/(x^2+5x+6), or even 7/x. The word rational comes from ratio: it is one polynomial divided by another, the same way a rational number is one integer divided by another.

Everything you already know about an ordinary fraction still applies. The bar means divide. You can reduce, multiply, add, and so on. The only genuinely new wrinkle is that the bottom is no longer a fixed number — it changes as the variable changes, and sometimes it lands on zero.

Division by zero is the one rule that matters

Dividing by zero has no meaning, so any input that makes the denominator zero is an excluded value: the expression simply does not exist there. Finding these is mechanical — set the denominator equal to zero and solve.

Find the excluded values of  (x + 2) / (x^2 + 5x + 6)

Set the denominator to zero:
  x^2 + 5x + 6 = 0
  (x + 2)(x + 3) = 0
  x = -2  or  x = -3

Excluded values:  x ≠ -2  and  x ≠ -3
The expression is defined for every other real number.

Excluded values and the domain

The list of forbidden inputs is exactly a domain restriction. When you treat the expression as a rational function f(x), its domain is all real numbers except those excluded values. So the same fact wears two names: an excluded value of the expression is a gap in the domain of the function.

1. Look only at the denominator; the numerator never causes an exclusion.
2. Set the denominator equal to zero and solve that equation completely.
3. Every solution you find is an excluded value; write each as x ≠ that number.
4. State the domain as all real numbers except those values.