Simplifying Rational Expressions where P(x) and Q(x) are polynomials, Q(x) ≠ 0. Just as we simplify rational numbers (fractions) in arithmetic, we want to be able to simplify rational expressions. Recall that a rational expression is given by

To simplify rational expressions: 1.Factor both the numerator and the denominator. 2.Reduce any common factors that are present in both the numerator and the denominator.

Example 1 Factor the numerator and the denominator. Simplify the rational expression

The simplified rational expression is … Reduce common factors.

The variable x found in both the numerator and the denominator cannot be reduced! Important note: The variable x in the numerator is a term, not a factor. Therefore, it cannot be reduced. The variable x in the denominator is a term within the factor (x + 2), and is not a factor itself. Therefore, it cannot be reduced.

Remember: we can only reduce common factors. The only time a term can be reduced is when it is a factor. This only happens when the term makes up the entire numerator or entire denominator.

Example 2 Since the term 3x is the whole numerator, it is a factor of the numerator, and we can reduce with the 3x factor in the denominator. Simplify the rational expression

When we reduce, we are actually removing a factor equal to 1. The reason behind the reducing is shown by the following:

Example 3 Simplify the rational function, and note any restrictions on the domain.

To determine the domain restrictions, find the values that will make the original denominator 0: Write your answer as follows:

Example 4 The 3x in the numerator is just a term now, and not a factor. Therefore we cannot reduce with the 3x in the numerator. Simplify the rational expression Since neither numerator nor denominator can be factored further, and there are no common factors, the expression is simplified.

Example 5 Simplify the rational expression Factor both numerator and denominator.

Reduce the common factors.

Example 6 Simplify the rational expression Since the x’s and 4’s are terms, they can’t be reduced. At first glance, one would think that the expression is simplified.

Factor a negative one out of the denominator. Reduce the common binomial factors to get …