Rational expressions are multiplied and divided.
After completing this tutorial, you will be able to complete the following:
Rational expressions are a type of algebraic expression with direct applications to such diverse areas as number theory and population modeling. The process of multiplying or dividing rational expressions is a foundational component of mathematical fluency on which a great deal of mathematical concepts rely.
Rational expressions are quotients of polynomial expressions. This definition is analogous to the definition of rational numbers as quotients of integers. This reasoning provides the loose beginnings of a dictionary between types of numbers and types of expressions.
The analogy can be extended to include real and complex numbers. Elaborating upon and making this dictionary precise has been a core area of mathematical research for the last 200 years.
Rational expressions can be multiplied or divided similarly to rational numbers. In what follows, we consider only rational expressions that are the quotient of polynomials in the single variable x. This will help to reduce the difficulty posed by complicated or confusing notation.
Let and be polynomials and assume that is not identically zero. The quotient
is a rational expression. The polynomial is called the numerator and the denominator. For example, the following two quotients are rational expressions:
Defined as such, the space of rational expressions possesses many of the same arithmetic properties as rational numbers. Rational expressions can be combined by addition, subtraction, multiplication, or division. These operations are conducted by using the familiar procedures from fraction arithmetic.
To multiply two rational expressions, one proceeds as if multiplying rational numbers. Recall, for example, how to multiply fractions.
The product is taken by multiplying straight across: numerator times numerator and denominator times denominator. In the initial stages, the process is nearly identical for rational expressions.
At this point, an important distinction emerges. In high or middle school math courses, x is commonly regarded as a free variable over the real numbers. In the above expression, however, the values 1 and are excluded from the domain of x because either number results in division by zero. Therefore, before simplifying the product, we first note the exclusions.
Next, we factor the numerator and cancel common factors.
The reason for excluding values and copying the information to each line is now clear. If x were a free variable over the real numbers, then x could take on the values of 1 or . The previous expressions are undefined for these values. Therefore, the expression x, interpreted as a function, is equivalent to only on domains that do not include 1 or . Additionally, this exclusion could not be inferred from the last line alone, since it is not written as a quotient.
To divide rational expressions, one follows the same method used to divide fractions, and then applies the previous reasoning about rational expressions. The following is an example:
This Activity Object introduces these methods for multiplying and dividing rational expressions by comparing them to rational numbers and demonstrating each step of the process.
|Approximate Time||30 Minutes|
|Pre-requisite Concepts||Students should understand how to multiply and divide rational numbers, and factor algebraic expressions.|
|Type of Tutorial||Skills Application|
|Key Vocabulary||dividing rational expressions, multiplying rational expressions, rational expression|