
A rational expression is an expression that is the ratio of two polynomials. 
Rational expressions are algebraic fractions in which the numerator is a polynomial and the denominator is also a polynomial (usually different from the numerator). The polynomials used in creating a rational expression may contain one term (monomial), two terms (binomial), three terms (trinomial), and so on.
Rational Expressions
(monomial/monomial) 
Rational Expression
(binomial/binomial) 
Rational Expression
(binomial/trinomial) 




Expressions that are not polynomials
cannot be used in the creation of
rational expressions. 
For example: is not a rational expression, since is not a polynomial.
Since rational expressions represent division, we must be careful to
avoid division by zero.
If a rational fraction has a variable in its denominator, we must ensure that any
value (or values) substituted for that variable will not create a zero denominator.
If it is not obvious which values will cause a division by zero error in a rational expression,
set the denominator equal to zero and solve for the variable.
Rational expression: Could it possibly
be undefined? When? 
Rational expression: Could it possibly
be undefined? When? 

Obviously, when x = 1, the denominator will be zero, making the expression undefined.
Domain: 

Set the denominator = 0
and solve.
a^{2}  4 = 0
a^{2} = 4

For this rational expression, we must limit the x's which
may be used, to avoid a division by zero error, and
leaving the expression undefined.

For this rational expression, we must prevent two
xvalues from being used in the expression.
Domain: 
Rational expression: Could it possibly
be undefined? When? 
Rational expression: Could it possibly
be undefined? When? 

Set: 8  y = 0
8 = y
Domain: All real numbers, except y = 8. 

Set: x^{2} + x  12 = 0
(x  3)(x + 4) = 0
x  3 = 0; x = 3
x + 4 = 0; x = 4
Domain: All real numbers, but not x = 3 and not x = 4. 