Simplifying Rational Expressions Recall that a rational expression
- Slides: 17
Simplifying Rational Expressions • Recall that a rational expression is given by 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. Table of Contents
• 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. Table of Contents
• Example 1 Simplify the rational expression Factor the numerator and the denominator. Table of Contents
Reduce common factors. The simplified rational expression is … Table of Contents
Important note: The variable x found in both the numerator and the denominator cannot be reduced! 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. Table of Contents
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. Table of Contents
• Example 2 Simplify the rational expression Since the term 3 x is the whole numerator, it is a factor of the numerator, and we can reduce with the 3 x factor in the denominator. Table of Contents
The reason behind the reducing is shown by the following: When we reduce, we are actually removing a factor equal to 1. Table of Contents
• Example 3 Simplify the rational function, and note any restrictions on the domain. Table of Contents
To determine the domain restrictions, find the values that will make the original denominator 0: Write your answer as follows: Table of Contents
• Example 4 Simplify the rational expression The 3 x in the numerator is just a term now, and not a factor. Therefore we cannot reduce with the 3 x in the numerator. Since neither numerator nor denominator can be factored further, and there are no common factors, the expression is simplified. Table of Contents
• Example 5 Simplify the rational expression Factor both numerator and denominator. Table of Contents
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Reduce the common factors. Table of Contents
• 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. Table of Contents
Factor a negative one out of the denominator. Reduce the common binomial factors to get … Table of Contents
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