What are Rational Expressions?
A rational expression is a ratio of two algebraic expressions. Specifically, we have a polynomial expression in the numerator, and a polynomial expression in the denominator of a fraction.
As functions, we can write a rational function as:
So we can refer to p(x) as the polynomial in the numerator, and q(x) as the polynomial in the denominator.
Here are some examples of rational expressions:
Domain of Rational Expressions
Because we have a denominator in rational expressions, we sometimes set restrictions to what the denominator can equal. We cannot divide by zero, otherwise the expression is undefined. Therefore, variables can take on any value, provided that it does not make the denominator equal zero. This represents a domain restriction.
Below, we are going to practice identifying domain restrictions by looking at rational expressions in different forms. In each example, we'll talk about how we can go about identifying x-values that make the denominator equal zero.
Identifying Domain Restrictions
Our first example is the expression:
To find domain restrictions, we focus on the denominator of the expression. The denominator is a polynomial expression in factored form. To find values of x which make the denominator equal to zero, we can set each factor equal to zero, and solve for x:
We can conclude that x is not allowed to be 1 or –2.
Next, let's consider a rational expression in which the polynomials are written in expanded form:
It would be ideal if we could factor the denominator, so that we can solve using a similar method. Notice that it is not necessary to factor the numerator at all, because we really do not care what the value of the numerator is. All that matters is that we find values for x that make the denominator equal to zero.
We notice that the denominator is a quadratic expression, so we will try to factor. If we cannot factor, using the quadratic formula is also an option. Fortunately, factoring the quadratic is possible, and it isn't terribly difficult. Since the constant term is positive, but the x-term coefficient is negative, we are looking for two negative integers that sum to –8, but multiply to 15. These happen to be –3 and –5:
We can conclude that x is not allowed to be 3 or 5.
In our last example, we have a rational expression with polynomials once again expressed in factored form. We also notice common factors in both the numerator and denominator that cancel out:
We might think that because this factor has been canceled away, we can ignore it completely. This is not true.
When cancelling out factors in rational expressions, it is important to still consider them when finding domain restrictions. We must also set the canceled out factors equal to zero and solve for x.
We can conclude that x is not allowed to be 2 or –3.
The denominator of rational expressions is not allowed to be zero. Therefore, we must set the denominator equal to zero, and solve for any x–values that make the denominator equal zero. If no such solutions exist, there are no restrictions to what x can be.
a fraction in which the numerator and denominator are polynomials