Rational Expressions and Rational Equations
1 Simplifying Rational Expressions
Rational functions are nothing more than fractions whose numerators and denominators are
polynomials. Because they are functions, they have domains. It may not be possible to
evaluate the function at any real number. Because rational functions are fractions, having
the denominator be zero gives us something which is undefined. This gives us the important
Important Idea. A rational expression is defined exactly where the denominator is not
Example 1. Determine the values of variable where each expression is defined.
Solution. The function is defined exactly where the
denominator is not zero. Therefore, we
find where the denominator is zero and eliminate those possibilities. The denominator will
be zero when
Using the quadratic formula, we have
a = 1, b = 4, c = -5.
The discriminant is then
The zeros are then given by
Thus, the domain of the expression will be all real numbers except 1 and -5.
Example 2. Determine the value of the variables where the rational expression is defined.
Solution. Again, the expression will be defined where the
denominator is not zero. Thus, we
need to solve
|(add 3 to both sides)|
|(divide both sides by 2)|
Thus the expression will be defined for all real numbers
Example 3. Determine the values of the variables where the denominator is defined.
Solution. The expression will be defined where the
denominator is not zero. We then find
where the denominator is zero and we eliminate those values. We therefore need to solve
Identifying the coefficients
a = 1, b = 1, c = 2.
Computing the discriminant, we have
no real solutions
Since the denominator is never zero, no values have to be
eliminated. This means the function
is defined for all real numbers.
Simplifying rational expressions is exactly like simplifying fractions. You factor the num-berator
and the denominator into primes and the cancel common factors between numerator
Example 4. Simplify
Solution. In the denominator, we can take out a factor of
3x + 9 = 3x + 3(3) = 3(x + 3).
Likewise, in the numerator we can take out a factor of 3 to give us
12x = 3(4x).
In our rational expression we then have
Thus, our simplified expression is